Visual understanding of centripetal acceleration formula

WEBVTT
Kind: captions
Language: en

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Let's say I have
some object that's

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traveling in a circular
path just like this.

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And what I've drawn here
is its velocity vector

00:00:07.490 --> 00:00:10.140
at different points
along that path.

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And so this right over
here is going to be v1,

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velocity vector 1.

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This is going to be
velocity vector 2.

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And this right over here is
going to be velocity vector 3.

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And what we're going to
assume, in this video,

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is that the magnitude of these
velocity vectors is constant.

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Or another way to think about it
is that the speed is constant.

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So I'll just say lowercase
v without the arrow on top--

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so this is going to be a scalar
quantity-- I'll call this

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the speed.

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Or you could call this the
magnitude of these vectors.

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And this is going
to be constant.

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So this is going to be equal
to the magnitude of vector

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1, which is equal to the
magnitude of vector 2.

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The direction is
clearly changing,

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but the magnitude is going
to be the same, which

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is equal to the
magnitude of vector 3.

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And we're going to assume
that it's traveling in a path,

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in a circle with radius r.

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And what I'm going
to do is, I'm going

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to draw a position
vector at each point.

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So let's call r1-- actually I'll
just do it in pink-- let's call

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r1 that right over there.

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That's position vector r1.

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That is position vector r2.

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So the position is
clearly changing.

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That's position vector r2.

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And that is position vector r3.

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But the magnitude of
our position vectors

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are clearly the same.

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And I'm going to call the
magnitude of our position

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vectors r.

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And that's just the
radius of the circle.

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It's this distance
right over here.

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So r is equal to
the magnitude of r1,

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which is equal to the
magnitude of r2, which

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is equal to the magnitude of r3.

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Now what I want to
do, in this video,

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is prove to you visually,
that given this radius

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and given this speed,
that the magnitude

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of the centripetal
acceleration-- and I'll

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just write that as a sub c,
I don't have an arrow on top,

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so this is a scalar quantity.

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So the magnitude of the
centripetal acceleration

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is going to be equal
to our speed squared,

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our constant speed
squared, divided

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by the radius of the circle.

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I want you to feel good
that this is indeed

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the case by the
end of this video.

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And to understand
that, what I want to do

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is I want to re-plot
these velocity

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vectors on another
circle and just

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think about how the vectors
themselves are changing.

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So let's copy and paste this.

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So let me copy and paste v1.

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So copy and paste.

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So that is v-- actually I
want to do it from the center

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--so that is v1.

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Then let me do the
same thing for v2.

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So let me copy and paste it.

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That is v2.

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And then let me
do it also for v3.

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I'll just get the vector part;
I don't have to get the label.

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So copy and paste it.

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And that right over
there is vector v3.

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And let me clean
this up a little bit.

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So that's clearly v2.

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I don't think we have
to label anymore.

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We know that v2 is in orange.

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And what is the
radius of this circle

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going to be right over here?

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Well, the radius
of this circle is

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going to be the magnitude
of the velocity vectors.

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And we already know the
magnitude of the velocity

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vectors is this quantity
v, this scalar quantity.

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So the radius of this circle is
v. The radius of this circle,

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we already know, is equal to r.

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And just as the
velocity vector is

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what's giving us the
change in position

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over time, the change in
position vector over time,

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what's the vector
that's going to give us

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the change in our
velocity vector over time?

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Well, that's going to be
our acceleration vectors.

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So you will have
some acceleration.

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We'll call this a1.

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We'll call this a2.

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And I'll call this a3.

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And I want to make
sure that you get

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the analogy that's
going on here.

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As we go around in this
circle, the position

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vectors first they point out
to the left, then the upper,

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kind of in a maybe
11 o'clock position,

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or I guess the top left,
and then to the top.

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So it's pointing in these
different directions

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like a hand in a clock.

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And what's moving it
along there is the change

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in position vector
over time, which

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are these velocity vectors.

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Over here, the velocity
vectors are moving around

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like the hands of a clock.

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And what is doing
the moving around

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are these acceleration vectors.

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And over here, the
velocity vectors

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are tangential to the
path, which is a circle.

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They're perpendicular
to a radius.

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And you learned
that in geometry--

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that a line that is
tangent to a circle

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is perpendicular to a radius.

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And it's also going to be the
same thing right over here.

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And just going back
to what we learned

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when we learned
about the intuition

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of centripetal acceleration, if
you look at a1 right over here,

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and you translate this vector,
it'll be going just like that.

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It is going towards the center.

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a2, once again, is going
towards the center.

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a3, once again, if
you translate that,

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that is going
towards the center.

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So all of these are actually
center-seeking vectors.

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And you see that
right over here.

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These are actually centripetal
acceleration vectors

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right over here.

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Here we're talking about
just the magnitude of it.

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And we're going to
assume that all of these

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have the same magnitude.

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So we're going to
assume that they all

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have a magnitude of
what we'll call a sub c.

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So that's the magnitude.

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And it's equal to
the magnitude of a1.

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That vector, it's equal
to the magnitude of a2.

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And it's equal to
the magnitude of a3.

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Now what I want to think
about is how long is it

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going to take for
this thing to get

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from this point on the circle
to that point on that circle

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right over there?

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So the way to think about it
is, what's the length of the arc

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that it traveled?

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The length of this arc that
it traveled right over there.

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That's 1/4 around the circle.

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It's going to be 1/4
of the circumference.

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The circumference is 2 pi r.

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It Is going to be 1/4 of that.

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So that is the
length of the arc.

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And then how long will
it take it to go that?

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Well, you would
divide the length

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of your path divided by the
actual speed, the actual thing

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that's nudging it
along that path.

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So you want to divide that by
the magnitude of your velocity,

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or your speed.

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This is the magnitude of
velocity, not velocity.

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This is not a vector right
over here, this is a scalar.

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So this is going to be the
time to travel along that path.

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Now the time to
travel along this path

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is going to be the exact
same amount of time

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it takes to travel along this
path for the velocity vector.

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So this is for the position
vector to travel like that.

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This is for the velocity
vector to travel like that.

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So it's going to be
the exact same T.

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And what is the
length of this path?

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And now think of it in the
purely geometrical sense.

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We're looking at a circle here.

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The radius of the circle
is v. So the length

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of this path right over
here is going to be 1/4.

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It is going to be-- I'll
do it in that same color

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so you see the analogy--
it's equal to 1/4 times

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the circumference of the circle.

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The circumference of
this circle is 2 pi times

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the radius of the
circle, which is v.

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Now what is nudging
it along this circle?

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What is nudging it
along this path?

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What is the analogy for
speed right over here?

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Speed is what's nudging it
along the path over here.

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It is the magnitude of
the velocity vector.

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So what's nudging it along
this arc right over here

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is the magnitude of the
acceleration vector.

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So it is going to be a sub c.

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And these times are going
to be the exact same thing.

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The amount of time it
takes for this vector

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to go like that, for
the position vector,

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is the same amount of
time it takes the velocity

00:08:06.670 --> 00:08:08.640
vector to go like that.

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So we can set these 2
things equal each other.

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So we get, on this side,
we get 1/4 2 pi r over v

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is equal to 1/4 2 pi v over the
magnitude of our acceleration

00:08:28.730 --> 00:08:29.900
vector.

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And now we can simplify
it a little bit.

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We can divide both sides by 1/4.

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Get rid of that.

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We can divide both sides
by 2 pi, get rid of that.

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Let me rewrite it.

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So then we get r/v
is equal to v over

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the centripetal acceleration.

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And now you can cross multiply.

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And so you get v
times v. So I'm just

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cross multiplying right
over here. v times v,

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you get v squared, is
equal to a c times r.

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And cross multiplying, remember,
is really just the same thing

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as multiplying both sides
by both denominators,

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by multiplying both
sides times v and ac.

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So it's not some magical thing.

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If you multiply both sides times
v and ac, these v's cancel out.

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These ac's cancel out.

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You get v times v is v squared,
is equal to a sub c times r.

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And now to solve
for the magnitude

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of our centripetal
acceleration, you

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just divide both sides by r.

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And you are left
with-- and I guess

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we've earned a drum
roll now-- the magnitude

00:09:35.740 --> 00:09:37.800
of our centripetal
acceleration is

00:09:37.800 --> 00:09:40.470
equal to the magnitude,
our constant magnitude

00:09:40.470 --> 00:09:41.410
of our velocity.

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So this right here
is our speed, divided

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by the radius of the circle.

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And we're done!


Centripetal force and acceleration intuition

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:02.440
Let's say we observe some object--

00:00:02.440 --> 00:00:04.970
let's say for the sake of argument, it's happening in space

00:00:05.070 --> 00:00:07.870
It's traveling in a circular path with

00:00:08.070 --> 00:00:11.100
the magnitude of its velocity being constant

00:00:11.130 --> 00:00:15.050
Let me draw its velocity vector

00:00:15.050 --> 00:00:18.510
The length of this arrow is the magnitude of the velocity

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I want to be clear. In order for it to be traveling in the circular path

00:00:21.700 --> 00:00:24.220
the direction of its velocity needs to be changing

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So this time the velocity vector might look like that

00:00:27.810 --> 00:00:36.880
After a few seconds the velocity vector might look like this

00:00:37.210 --> 00:00:41.210
After another few seconds the velocity vector might look like this

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I'm just sampling. I actually could've sampled after a less time and it would be right over there

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but I am just sampling sometimes as it travels around the circle

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After a few more seconds the velocity vector might look something like that

00:00:57.740 --> 00:01:01.120
I want to think about what needs to happen

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what kind of force would have to act

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in particular the direction of the force would have to act on this object

00:01:07.000 --> 00:01:09.500
in order for the velocity vector to change like that?

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This remind ourselves if there was no force acting on this body

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this comes straight from Newton's 1st Law of motion

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then the velocity would not change

00:01:19.930 --> 00:01:23.450
neither the magnitude nor the direction of the velocity will change

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If there were no force acting on the subject

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it would just continue going on in the direction it was going

00:01:28.940 --> 00:01:34.230
it wouldn't curve; it wouldn't turn; the direction of its velocity wasn't changing

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Let's think about what the direction of that force would have to be

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and to do that, I'm gonna copy and paste these velocity vectors and keep

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track of what the direction of the change in velocity has to be

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Copy and paste that

00:01:52.350 --> 00:02:00.210
So that is our first velocity vector

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Copy all of these. This is our second one right over here

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Copy and paste it

00:02:10.900 --> 00:02:14.210
I'm just looking at it from the object's point of view

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how does the velocity vector change from each of these points in time to the next?

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Let me get all of these in there

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This green one

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That. Copy and paste it

00:02:31.010 --> 00:02:36.890
That. I could keep going, keep drawing velocity vectors around the circle

00:02:36.890 --> 00:02:39.530
but let me do this orange one right over here

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Copy and paste

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So between this magenta time and this purple time

00:02:57.060 --> 00:02:59.160
what was the change in velocity?

00:02:59.160 --> 00:03:01.950
Well, we could look at that purely from these vectors right here

00:03:01.950 --> 00:03:06.110
The change in velocity between those two times was that right over there

00:03:06.490 --> 00:03:09.110
That is our change in velocity

00:03:09.490 --> 00:03:15.900
So I take this vector and say in what direction was the velocity changing

00:03:15.900 --> 00:03:18.440
when this vector was going on this part of the arc

00:03:18.700 --> 00:03:22.640
It's roughly--if I just translate that vector right over here

00:03:22.770 --> 00:03:24.850
it's roughly going in that direction

00:03:24.930 --> 00:03:28.010
So that is the direction of our change in velocity

00:03:28.010 --> 00:03:31.160
This triangle is delta; delta is for change

00:03:31.570 --> 00:03:33.970
Now think about the next time period

00:03:34.630 --> 00:03:38.020
between this blue or purple period and this green period

00:03:38.020 --> 00:03:41.750
Our change in velocity would look like that

00:03:41.970 --> 00:03:44.260
So while it's traveling along this part of the arc

00:03:44.260 --> 00:03:49.180
roughly it's the change in velocity if we draw the vector starting at the object

00:03:49.190 --> 00:03:54.710
It would look something like this

00:03:55.020 --> 00:03:58.050
I'm just translating this vector right over here

00:03:58.290 --> 00:04:00.310
I'll do it one more time

00:04:00.340 --> 00:04:05.500
From this green point in time to this orange point in time

00:04:05.500 --> 00:04:08.520
and obviously we're just sampling points continuously moving

00:04:08.530 --> 00:04:10.980
and the change in velocity actually continues changing

00:04:10.980 --> 00:04:12.770
but hopefully you're going to see the pattern here

00:04:13.130 --> 00:04:21.340
So between those two points in time, this is our change in velocity

00:04:21.340 --> 00:04:23.730
And let me translate that vector right over there

00:04:23.780 --> 00:04:25.860
It would look something like that

00:04:25.880 --> 00:04:27.590
change in velocity

00:04:27.600 --> 00:04:32.330
So what do you see, if I were to keep drawing more of these change in velocity vectors

00:04:32.330 --> 00:04:35.330
you would see at this point, the change in velocity would have to be going

00:04:35.330 --> 00:04:36.590
generally in that direction

00:04:36.710 --> 00:04:40.920
At this point, the change in velocity would have to be going generally in that direction

00:04:41.140 --> 00:04:46.490
So what do you see? What's the pattern for any point along this circular curve?

00:04:46.780 --> 00:04:48.970
Well, the change in velocity

00:04:48.970 --> 00:04:53.340
first of all, is perpendicular to the direction of the velocity itself

00:04:53.840 --> 00:04:57.040
And we haven't proved it, but it at least looks like it

00:04:57.060 --> 00:04:59.080
Looks like this is perpendicular

00:04:59.110 --> 00:05:02.870
And even more interesting, it looks like it's seeking the center

00:05:02.870 --> 00:05:10.410
The change in velocity is constantly going in the direction of the center of our circle

00:05:11.110 --> 00:05:13.560
And we know from Newton's first law

00:05:13.760 --> 00:05:17.620
that if--the magnitude could stay the same but the

00:05:17.620 --> 00:05:21.710
velocity change in any way, either the magnitude or the direction or both

00:05:21.840 --> 00:05:25.430
there must be a net force acting on the object

00:05:25.440 --> 00:05:28.790
And the net force is acting in the direction of the acceleration

00:05:28.790 --> 00:05:30.770
which is causing the change in velocity

00:05:30.800 --> 00:05:34.450
So the force must be acting in the same direction as this change in velocity

00:05:34.470 --> 00:05:37.950
So in order make this object go in this circular

00:05:38.160 --> 00:05:44.710
there must be some force kind of pulling the object towards the center

00:05:44.730 --> 00:05:48.530
and a force that is perpendicular to its directional motion

00:05:48.840 --> 00:05:52.300
And this force is called the centripetal force

00:05:52.560 --> 00:05:55.850
Centripetal

00:05:55.850 --> 00:05:59.490
Not to be confused with centrifugal force, very different

00:05:59.600 --> 00:06:04.400
Centripetal force, centri- you might recognize as center

00:06:04.430 --> 00:06:13.910
and then -petal is seeking the center. It is center seeking

00:06:14.220 --> 00:06:18.650
So this centripetal force, something is pulling on this object towards the center that

00:06:18.680 --> 00:06:21.340
causes it to go into this circular motion

00:06:21.490 --> 00:06:25.350
Inward pulling causes inward acceleration

00:06:25.350 --> 00:06:28.730
So that's centripetal force

00:06:29.040 --> 00:06:34.190
causing centripetal acceleration

00:06:34.230 --> 00:06:37.130
which causes the object to go towards the center

00:06:37.130 --> 00:06:40.580
The whole point why I did this is that at least it wasn't intuitive to me

00:06:40.580 --> 00:06:42.760
that if you have this object going in a circle

00:06:42.800 --> 00:06:47.220
that the change in velocity, the acceleration, the force acting on this object

00:06:47.240 --> 00:06:51.550
would actually have to be towards the center

00:06:51.570 --> 00:06:53.780
The whole reason why I drew these vectors

00:06:53.790 --> 00:06:57.220
and then translate them over here and drew these change in velocity vectors

00:06:57.340 --> 00:07:04.830
is to show you that the change in velocity is actually towards the center of this circle

00:07:05.280 --> 00:07:09.600
Now with that out of the way, you might say, well, where is this happening in in everyday life

00:07:09.600 --> 00:07:13.240
or in reality in some way it perform

00:07:13.300 --> 00:07:16.880
And the most typical example of this and this is something that I think most of us have done

00:07:16.880 --> 00:07:20.900
when we were kid if you had a yoyo

00:07:21.170 --> 00:07:23.550
My best attempt to draw a yoyo

00:07:24.020 --> 00:07:29.680
If you have a yoyo and if you whip it around on a string

00:07:29.940 --> 00:07:36.330
you know that the yoyo goes in a circle

00:07:36.350 --> 00:07:42.080
Even though its speed might be constant, or the magnitude of its velocity might be constant

00:07:42.200 --> 00:07:45.090
we know that the direction of its velocity is constantly changing

00:07:45.090 --> 00:07:49.830
It's going in a circle and what's causing it to go in a circle is your hand right over here

00:07:50.010 --> 00:07:54.200
pulling on this string and providing tension into the string

00:07:54.490 --> 00:07:59.330
So there's a force, the centripetal force in this yoyo example is the tension in the string

00:07:59.480 --> 00:08:03.180
that's constantly pulling on the yoyo towards the center

00:08:03.180 --> 00:08:05.720
and that's why that yoyo goes in a circle

00:08:05.730 --> 00:08:10.090
Another example that you are probably somewhat familiar with or at least have heard about

00:08:10.090 --> 00:08:13.380
is if you have something in orbit around the planet

00:08:13.380 --> 00:08:16.630
So let's say this is Earth right here

00:08:16.670 --> 00:08:27.620
and you have some type of a satellite that is in orbit around Earth

00:08:27.790 --> 00:08:33.140
That satellite has some velocity at any given moment in time

00:08:33.260 --> 00:08:37.740
What's keeping it from not flying out into space and keeping it going in a circle

00:08:37.770 --> 00:08:39.590
is the force of gravity

00:08:39.590 --> 00:08:43.750
So in the example of a satellite or anything in the orbit

00:08:43.750 --> 00:08:46.190
even the moon in orbit around the Earth

00:08:46.500 --> 00:08:49.830
the thing that's keeping an orbit as opposed to flying out into space

00:08:49.890 --> 00:08:53.430
is a centripetal force of Earth's gravity

00:08:53.650 --> 00:08:58.010
Now another example, this is probably the most everyday example because we do it all the time

00:08:58.480 --> 00:09:01.860
If you imagine a car traveling around the racetrack

00:09:01.860 --> 00:09:05.860
Let's draw a racetrack. If I have a racetrack

00:09:06.330 --> 00:09:10.250
Before I tell you the answer, I'll have you think about it

00:09:10.900 --> 00:09:15.190
It's circular. Let's view the racetrack from above

00:09:15.260 --> 00:09:19.470
If I have a car on a racetrack. I want you to pause it before I tell it to you

00:09:19.470 --> 00:09:21.700
because I think it's an interesting thing think about

00:09:21.730 --> 00:09:24.910
It seems like a very obvious thing that's happening

00:09:24.910 --> 00:09:28.040
We've all experienced; we've all taken turns in cars

00:09:28.040 --> 00:09:32.810
So we're looking at the top of a car. Tires

00:09:32.830 --> 00:09:35.650
When you see a car going at a constant speed

00:09:35.800 --> 00:09:40.560
so on the speedometer, it might say, 60 mph, 40 mph, whatever the constant speed

00:09:40.560 --> 00:09:44.530
but it's traveling in a circle

00:09:44.810 --> 00:09:48.840
so what is keeping--what is the centripetal force in that example?

00:09:48.840 --> 00:09:52.340
There's no obvious string being pulled on the car towards the center

00:09:52.470 --> 00:09:56.660
There is no some magical gravity pulling it towards the center of the circle

00:09:56.660 --> 00:09:59.170
There's obviously gravity pulling you down towards the ground

00:09:59.180 --> 00:10:01.280
but nothing pulling it to the side like this

00:10:01.430 --> 00:10:07.180
So what's causing this car to go in the circle as opposed to going straight?

00:10:07.190 --> 00:10:10.390
And I encourage you to pause it right now before I tell you the answer

00:10:10.420 --> 00:10:13.520
Assuming you now unpaused it and I will now tell you the answer

00:10:13.670 --> 00:10:17.370
The thing that's keeping it going in the circle is actually the force of friction

00:10:17.660 --> 00:10:26.750
It's actually the force between the resist movement to the side between the tires and the road

00:10:26.900 --> 00:10:30.270
And a good example of that is if you would remove the friction

00:10:30.270 --> 00:10:33.770
if you would make the car driving on oil or on ice

00:10:33.800 --> 00:10:39.070
or if you would shave the treads of the tire or something

00:10:39.070 --> 00:10:40.990
then the car would not be able to do this

00:10:41.000 --> 00:10:43.240
So it's actually the force of friction in this example

00:10:43.250 --> 00:10:45.560
I encourage you to think about that


Negative and positive exponents

WEBVTT
Kind: captions
Language: en

00:00:00.910 --> 00:00:03.270
Rewrite the following
expression as the product

00:00:03.270 --> 00:00:06.380
of positive exponents, and
then evaluate the expression

00:00:06.380 --> 00:00:07.710
when x is equal to 2.

00:00:07.710 --> 00:00:10.030
So they gave us x to
the negative 3 times

00:00:10.030 --> 00:00:13.090
5 to the negative
2 times x squared.

00:00:13.090 --> 00:00:14.660
So the main thing
to realize here

00:00:14.660 --> 00:00:19.460
is if I have something
to a negative exponent,

00:00:19.460 --> 00:00:21.660
this is the exact
same thing as 1

00:00:21.660 --> 00:00:25.100
over that same base to
the positive version

00:00:25.100 --> 00:00:25.810
of the exponent.

00:00:25.810 --> 00:00:28.910
So x to the negative a is
equal to 1 over x to the a.

00:00:28.910 --> 00:00:32.060
So using this knowledge, let's
just apply it right over here.

00:00:32.060 --> 00:00:36.410
So x to the negative 3
power, we can rewrite as 1

00:00:36.410 --> 00:00:39.130
over x to the third power.

00:00:39.130 --> 00:00:41.980
And then 5 to the
negative 2, we can

00:00:41.980 --> 00:00:45.650
rewrite as 1 over 5 squared.

00:00:45.650 --> 00:00:50.280
And then finally, we
have times x squared.

00:00:50.280 --> 00:00:51.670
And so we can multiply this out.

00:00:51.670 --> 00:00:53.390
In the numerator, we
have 1 times 1 times

00:00:53.390 --> 00:00:56.580
x squared, which is
just equal to x squared.

00:00:56.580 --> 00:01:00.060
In the denominator, we have x
to the third times 5 squared.

00:01:00.060 --> 00:01:02.350
5 squared is 25.

00:01:02.350 --> 00:01:05.390
And here, you could say
implicitly there is a 1.

00:01:05.390 --> 00:01:07.360
This is the same thing
as x squared over 1.

00:01:07.360 --> 00:01:11.130
So in the denominator, we
have 25 times x to the third.

00:01:11.130 --> 00:01:15.810
So this is going to be
25 times x to the third.

00:01:15.810 --> 00:01:17.524
And then we can
simplify this even more

00:01:17.524 --> 00:01:19.440
because both the numerator
and the denominator

00:01:19.440 --> 00:01:21.690
is divisible by x squared.

00:01:21.690 --> 00:01:24.860
x squared divided
by x squared is 1.

00:01:24.860 --> 00:01:27.000
x to the third
divided by x squared

00:01:27.000 --> 00:01:29.660
is just going to be equal to x.

00:01:29.660 --> 00:01:31.610
And you could use other
exponent properties.

00:01:31.610 --> 00:01:33.630
You could say x squared
over x to the third

00:01:33.630 --> 00:01:37.070
is the same thing as x to
the 2 minus 3 power, which

00:01:37.070 --> 00:01:38.950
is x to the negative 1 power.

00:01:38.950 --> 00:01:40.800
But x to the negative
1 power, we know

00:01:40.800 --> 00:01:44.240
is the same thing as 1/x.

00:01:44.240 --> 00:01:47.390
Any way you do it, as long as
you do it in a reasonable way,

00:01:47.390 --> 00:01:49.290
you should get the same result.

00:01:49.290 --> 00:01:50.920
So when we simplify
it, we get 1/25x.

00:01:55.920 --> 00:02:00.980
Now, they want us to evaluate
when x is equal to 2.

00:02:00.980 --> 00:02:04.290
So if x is equal to 2,
we get 1 over 25 times 2,

00:02:04.290 --> 00:02:06.840
which is equal to 1/50.

00:02:06.840 --> 00:02:08.180
And we are done.

00:02:08.180 --> 00:02:11.792
We rewrote the expression as a
product of positive exponents.

00:02:11.792 --> 00:02:12.750
That's right over here.

00:02:12.750 --> 00:02:14.640
We made all the
exponents positive.

00:02:14.640 --> 00:02:17.690
And then we evaluated,
and we got 1/50.


Rational number word problem with decimals

WEBVTT
Kind: captions
Language: en

00:00:00.700 --> 00:00:03.250
At the beginning of the week,
Stewart's checking account

00:00:03.250 --> 00:00:07.530
had a balance of
negative $15.08.

00:00:07.530 --> 00:00:12.740
On Monday morning, he
deposited a check for $426.90.

00:00:12.740 --> 00:00:17.350
On Tuesday morning, he deposited
another check for $100.

00:00:17.350 --> 00:00:19.830
How much was in Stewart's
checking account

00:00:19.830 --> 00:00:23.790
after the second deposit, so
after both of these deposits

00:00:23.790 --> 00:00:25.080
right over here.

00:00:25.080 --> 00:00:27.130
So he starts off with
a negative balance.

00:00:27.130 --> 00:00:28.920
So a negative balance
means that he's

00:00:28.920 --> 00:00:30.720
overdrawn his checking account.

00:00:30.720 --> 00:00:32.997
He actually owes
the bank money now.

00:00:32.997 --> 00:00:35.580
Luckily, he's now going to put
some money in his bank account.

00:00:35.580 --> 00:00:37.340
So he'll actually have
a positive balance

00:00:37.340 --> 00:00:38.810
in his checking account.

00:00:38.810 --> 00:00:43.730
So he's starts off with
the negative $15.08.

00:00:43.730 --> 00:00:50.130
And then to that,
he adds $426.90.

00:00:50.130 --> 00:00:55.260
And then he adds another $100.

00:00:55.260 --> 00:00:59.010
So he started off
with negative $15.08.

00:00:59.010 --> 00:01:04.810
And then to that, he
adds $426.90 and $100.

00:01:04.810 --> 00:01:09.272
So that's $526.90.

00:01:09.272 --> 00:01:11.480
And so how much is going to
have in his bank account?

00:01:11.480 --> 00:01:18.540
He started owing $15.08, and
then he's going to add $526.90.

00:01:18.540 --> 00:01:21.940
So one way to visualize
it is, if you think

00:01:21.940 --> 00:01:27.090
about it on a number line,
if this is 0 right over here,

00:01:27.090 --> 00:01:30.860
he's going to start
off at negative $15.08.

00:01:30.860 --> 00:01:35.060
But then he's going to add $526.

00:01:35.060 --> 00:01:40.840
So this right over here,
this is $15.08 to the left.

00:01:40.840 --> 00:01:42.200
That's how much he owes.

00:01:42.200 --> 00:01:45.680
And to that, he's
going to add $526.

00:01:45.680 --> 00:01:47.470
So I'm not drawing
this to scale.

00:01:47.470 --> 00:01:53.861
But to that, he is
going to add $526.90.

00:01:53.861 --> 00:01:56.360
So the amount that he's going
to be in the positive is going

00:01:56.360 --> 00:02:01.940
to be $526.90 minus the $15.08.

00:02:01.940 --> 00:02:07.209
It's essentially going to be
this length right over here.

00:02:07.209 --> 00:02:09.250
That's how much he's going
to be in the positive.

00:02:09.250 --> 00:02:16.730
And that's going to be
$526.90 minus $15.08.

00:02:16.730 --> 00:02:20.630
So that's going to be, and
we can even just rewrite this

00:02:20.630 --> 00:02:22.460
so it actually looks
exactly like that.

00:02:22.460 --> 00:02:29.307
That's exactly the same
thing as $526.90 minus--

00:02:29.307 --> 00:02:31.390
adding a negative is the
same thing as subtracting

00:02:31.390 --> 00:02:34.320
a positive-- minus $15.08.

00:02:34.320 --> 00:02:37.000
And this is-- I will do
this in another color--

00:02:37.000 --> 00:02:44.290
$526.90 minus $15.08.

00:02:44.290 --> 00:02:46.300
Let's see, 0 is less than 8.

00:02:46.300 --> 00:02:49.110
Let's make that a 10
and borrow from this 9.

00:02:49.110 --> 00:02:51.820
So that becomes an 8, or I guess
you could say we're regrouping.

00:02:51.820 --> 00:02:54.290
Now, everything up here is
larger than everything there.

00:02:54.290 --> 00:02:56.380
So 10 minus 8 is 2.

00:02:56.380 --> 00:02:58.190
8 minus 0 is 8.

00:02:58.190 --> 00:02:59.570
We have our decimal.

00:02:59.570 --> 00:03:01.890
6 minus 5 is 1.

00:03:01.890 --> 00:03:03.695
2 minus 1 is 1.

00:03:03.695 --> 00:03:06.200
And then you have
5 minus nothing.

00:03:06.200 --> 00:03:13.734
So he is left with $511.82
after his second deposit.


Difference of cubes factoring

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:04.820
We're asked to factor 40c
to the third power minus 5d

00:00:04.820 --> 00:00:06.324
to the third power.

00:00:06.324 --> 00:00:08.199
So the first thing that
might jump out at you

00:00:08.199 --> 00:00:10.770
is that 5 is a factor
of both of these terms.

00:00:10.770 --> 00:00:16.470
I could rewrite this as 5
times 8c to the third power

00:00:16.470 --> 00:00:20.890
minus 5 times d to
the third power.

00:00:20.890 --> 00:00:23.440
And so you could actually
factor out a 5 here,

00:00:23.440 --> 00:00:26.560
so factor out a 5.

00:00:26.560 --> 00:00:28.200
And so if you
factor out a 5, you

00:00:28.200 --> 00:00:32.770
get 5 times 8c to
the third power

00:00:32.770 --> 00:00:35.670
minus d to the third power.

00:00:35.670 --> 00:00:37.150
So as you see,
factoring, it really

00:00:37.150 --> 00:00:40.180
is just undistributing
the 5, reversing

00:00:40.180 --> 00:00:41.930
the distributive property.

00:00:41.930 --> 00:00:43.750
And when you write
it like this, it

00:00:43.750 --> 00:00:46.190
might jump out at you
that 8 is a perfect cube.

00:00:46.190 --> 00:00:48.740
It's 2 to the third power.
c to the third power

00:00:48.740 --> 00:00:50.760
is obviously c to
the third power.

00:00:50.760 --> 00:00:52.477
And then you have d
to the third power.

00:00:52.477 --> 00:00:54.310
So this right here is
a difference of cubes.

00:00:54.310 --> 00:00:56.200
And actually, let me
write that explicitly

00:00:56.200 --> 00:00:59.550
because 8 is the same thing
as 2 to the third power.

00:00:59.550 --> 00:01:04.150
So you can write this as-- let
me write the 5 out front-- 5

00:01:04.150 --> 00:01:06.850
times-- this term
right over here

00:01:06.850 --> 00:01:11.322
can be rewritten as
2c to the third power

00:01:11.322 --> 00:01:13.030
because it's 2 to the
third power times c

00:01:13.030 --> 00:01:16.300
to the third power-- 8c to
the third power-- And then,

00:01:16.300 --> 00:01:18.580
minus d to the third power.

00:01:21.580 --> 00:01:25.980
And so this gives us, right over
here, a difference of cubes.

00:01:25.980 --> 00:01:31.210
And you can actually factor
a difference of cubes.

00:01:31.210 --> 00:01:33.120
And you may or may
not know the pattern.

00:01:33.120 --> 00:01:36.840
So if I have a to the
third minus b to the third,

00:01:36.840 --> 00:01:40.390
this can be factored
as a minus b

00:01:40.390 --> 00:01:45.980
times a squared plus
ab plus b squared.

00:01:45.980 --> 00:01:48.530
And if you don't believe me, I
encourage you to multiply this

00:01:48.530 --> 00:01:51.302
out, and you will get a to the
third minus b to the third.

00:01:51.302 --> 00:01:53.010
You get a bunch of
terms that cancel out,

00:01:53.010 --> 00:01:54.940
so you're only left
with two terms.

00:01:54.940 --> 00:01:56.870
And even though it's
not applicable here,

00:01:56.870 --> 00:02:01.120
it's also good to know
that the sum of cubes

00:02:01.120 --> 00:02:02.620
is also factorable.

00:02:02.620 --> 00:02:10.130
It's factorable as a plus b
times a squared minus ab plus b

00:02:10.130 --> 00:02:11.020
squared.

00:02:11.020 --> 00:02:12.970
So once again, I won't go
through the time of multiplying

00:02:12.970 --> 00:02:14.594
this out, but I
encourage you to do so.

00:02:14.594 --> 00:02:17.610
It just takes a little bit
of polynomial multiplication.

00:02:17.610 --> 00:02:19.750
And you'll be able
to prove to yourself

00:02:19.750 --> 00:02:21.490
that this is indeed the case.

00:02:21.490 --> 00:02:24.550
Now, assuming that
this is the case,

00:02:24.550 --> 00:02:27.150
we can just do a little
bit of pattern matching.

00:02:27.150 --> 00:02:35.815
Because in this case, our
a is 2c, and our b is d.

00:02:35.815 --> 00:02:44.530
So let me write this. a is equal
to 2c, and our b is equal to d.

00:02:44.530 --> 00:02:47.010
We have minus b to the third
and minus d to the third,

00:02:47.010 --> 00:02:49.400
so b and d must
be the same thing.

00:02:49.400 --> 00:02:51.690
So this part inside
must factor out

00:02:51.690 --> 00:02:55.650
to-- let me write my
5, open parentheses.

00:02:55.650 --> 00:02:58.140
Let me give myself some space.

00:02:58.140 --> 00:03:02.290
So it's going to factor
out into a minus b.

00:03:02.290 --> 00:03:08.480
So a is 2c minus b, which is d.

00:03:08.480 --> 00:03:12.500
So it factors out as the
difference of the two things

00:03:12.500 --> 00:03:14.190
that I'm taking the cube of.

00:03:14.190 --> 00:03:23.360
2c minus d times-- and now, I
have a squared is 2c squared.

00:03:23.360 --> 00:03:28.250
2c squared is the same
thing as 4c squared.

00:03:28.250 --> 00:03:32.000
Let me make that. a squared is
equal to 2c-- the whole thing

00:03:32.000 --> 00:03:34.490
squared, which is
equal to 4c squared.

00:03:34.490 --> 00:03:39.860
So it's 4c squared
plus a times b.

00:03:39.860 --> 00:03:47.570
So that's going to be 2c
times d, so plus 2c times d.

00:03:47.570 --> 00:03:51.670
And then finally plus b squared,
and in our case, b is d.

00:03:51.670 --> 00:03:54.564
So you get plus d squared.

00:03:54.564 --> 00:03:55.230
And you're done.

00:03:55.230 --> 00:03:56.380
We have factored it out.

00:03:56.380 --> 00:03:58.860
And actually, you could get
rid of one set of parentheses.

00:03:58.860 --> 00:04:04.990
This can be factored as 5
times 2c minus d times 4c

00:04:04.990 --> 00:04:09.380
squared plus 2cd plus d squared.

00:04:09.380 --> 00:04:12.040
And we are done.


Solutions to three variable system 2

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.190
Determine whether the
system has no solutions

00:00:03.190 --> 00:00:05.850
or infinite solutions.

00:00:05.850 --> 00:00:08.980
So let's think about how
we can go about doing this.

00:00:08.980 --> 00:00:12.790
So if at any point we might
not have to solve this entirely

00:00:12.790 --> 00:00:14.920
if we somehow get something
that's nonsensical

00:00:14.920 --> 00:00:16.700
which will tells us
there's no solutions.

00:00:16.700 --> 00:00:18.350
Or we might have to
go further and see

00:00:18.350 --> 00:00:19.810
if it's one or
infinite solutions,

00:00:19.810 --> 00:00:21.935
although it looks like one
solution isn't an option

00:00:21.935 --> 00:00:23.990
here, given how this
question is phrased.

00:00:23.990 --> 00:00:25.700
So the way that
you would proceed

00:00:25.700 --> 00:00:28.790
to solve three equations
with three unknowns

00:00:28.790 --> 00:00:31.990
is you would try to eliminate
variables one by one.

00:00:31.990 --> 00:00:34.364
And so first we could try to
eliminate the x variables.

00:00:34.364 --> 00:00:36.030
And we could do that,
we can essentially

00:00:36.030 --> 00:00:38.440
create two equations
with two unknowns.

00:00:38.440 --> 00:00:40.360
The two unknowns
will be y and z.

00:00:40.360 --> 00:00:42.890
If we can pair up
these equations

00:00:42.890 --> 00:00:45.420
and eliminate x with
each of these pairings.

00:00:45.420 --> 00:00:48.210
So for example, we can
pair these first two.

00:00:48.210 --> 00:00:52.040
We can pair the last two.

00:00:52.040 --> 00:00:55.680
And that's all we would need
to have to eliminate the x's

00:00:55.680 --> 00:00:56.990
and still have two equations.

00:00:56.990 --> 00:00:59.630
And have all of the information
of these three equations.

00:00:59.630 --> 00:01:01.088
But then the third
pairing would be

00:01:01.088 --> 00:01:02.930
the first and the
third equation,

00:01:02.930 --> 00:01:05.129
But we only have to do
two of these pairings.

00:01:05.129 --> 00:01:08.280
Now, just to show you what I
mean by these pairings, what

00:01:08.280 --> 00:01:11.062
I want to do is take
these first two.

00:01:11.062 --> 00:01:13.270
I'm going to pair this first
pairing right over here,

00:01:13.270 --> 00:01:16.910
and I'm going to use them
to eliminate the x terms.

00:01:16.910 --> 00:01:19.370
And over here I have
2x, over here I have 8x.

00:01:19.370 --> 00:01:21.760
If I could turn this
2x into a negative 8x

00:01:21.760 --> 00:01:24.310
I could add both sides of
these equations to each other

00:01:24.310 --> 00:01:26.510
and the x terms
would cancel out.

00:01:26.510 --> 00:01:31.020
And so the best way to turn
this 2x into a negative 8x

00:01:31.020 --> 00:01:36.150
is to multiply this top
equation times negative 4.

00:01:36.150 --> 00:01:38.650
When I say multiply, I'm saying
multiply the whole equation,

00:01:38.650 --> 00:01:41.220
both sides of it, by negative 4.

00:01:41.220 --> 00:01:44.640
So 2x times negative
4 is negative 8x.

00:01:44.640 --> 00:01:50.060
Negative 4y times negative 4
is plus 16y, are positive 16y.

00:01:50.060 --> 00:01:55.120
z times negative 4 is negative
4z And that's equal to 3 times

00:01:55.120 --> 00:01:58.460
negative 4 which is negative 12.

00:01:58.460 --> 00:02:00.950
And then I can rewrite this
equation right over here.

00:02:00.950 --> 00:02:08.100
It's 8x minus 2y plus
4z is equal to 7.

00:02:08.100 --> 00:02:11.630
And now I can add
these both equations.

00:02:11.630 --> 00:02:14.170
On the left hand side,
these guys cancel out,

00:02:14.170 --> 00:02:16.500
16y minus 2y-- and that
was the whole point

00:02:16.500 --> 00:02:19.610
behind multiplying the top
equation by negative 4--

00:02:19.610 --> 00:02:22.760
16y minus 2y is 14y.

00:02:22.760 --> 00:02:24.670
Negative 4z plus 4z.

00:02:24.670 --> 00:02:27.010
These guys actually
cancel out as well.

00:02:27.010 --> 00:02:28.700
So actually with
that one pairing,

00:02:28.700 --> 00:02:30.574
by multiplying by negative
4 we were actually

00:02:30.574 --> 00:02:32.090
able to cancel
out two variables.

00:02:32.090 --> 00:02:36.370
So you get 14y is
equal to negative 12

00:02:36.370 --> 00:02:39.500
plus 7 is equal to negative 5.

00:02:39.500 --> 00:02:41.590
And you can actually
solve for y.

00:02:41.590 --> 00:02:44.970
And we don't know if this one
will actually have solutions.

00:02:44.970 --> 00:02:47.242
But if we assume it's
going to have a solution,

00:02:47.242 --> 00:02:49.200
you could actually solve
for y right over here.

00:02:49.200 --> 00:02:50.860
You could divide
both sides by 14.

00:02:50.860 --> 00:02:53.130
But let's worry about
that a little bit later.

00:02:53.130 --> 00:02:55.442
Let's take the second
pairing right over here.

00:02:55.442 --> 00:02:56.650
So once again you have an 8x.

00:02:56.650 --> 00:02:58.630
We want to eliminate the x's.

00:02:58.630 --> 00:03:01.240
So this one you have an 8x,
here you have a negative 4x.

00:03:01.240 --> 00:03:03.325
If you multiply
this times 2, this

00:03:03.325 --> 00:03:04.950
is going to become
a negative 8x and it

00:03:04.950 --> 00:03:06.770
can cancel with this top one.

00:03:06.770 --> 00:03:12.174
So the top equation is 8x
minus 2y plus 4z is equal to 7.

00:03:12.174 --> 00:03:14.590
When I say "top equation" I'm
talking about this one right

00:03:14.590 --> 00:03:17.530
over here, the top
one in this pairing.

00:03:17.530 --> 00:03:19.590
And this bottom
equation, I'm going

00:03:19.590 --> 00:03:23.040
to multiply times negative 2.

00:03:23.040 --> 00:03:25.820
I'm going to multiply it
times negative 2-- sorry times

00:03:25.820 --> 00:03:27.280
positive 2.

00:03:27.280 --> 00:03:29.760
I'm going to multiply
it times positive 2.

00:03:29.760 --> 00:03:33.100
Negative 4x times
positive 2 is negative 8x.

00:03:33.100 --> 00:03:35.520
So I'm going to
multiply it by 2.

00:03:35.520 --> 00:03:40.030
So 2 times negative
4x is negative 8x.

00:03:40.030 --> 00:03:44.140
2 times y is plus 2y.

00:03:44.140 --> 00:03:47.850
2 times negative
2z is negative 4z.

00:03:47.850 --> 00:03:51.787
And then 2 times negative
14 is negative 28.

00:03:51.787 --> 00:03:53.370
And now we can add
the left hand sides

00:03:53.370 --> 00:03:55.080
and add the right hand sides.

00:03:55.080 --> 00:03:58.180
These cancel out, those
cancel out, those cancel out.

00:03:58.180 --> 00:04:00.430
We actually end up with
nothing on the left hand side.

00:04:00.430 --> 00:04:02.480
You get 0 plus 0 plus 0.

00:04:02.480 --> 00:04:05.630
And on the right hand side
you get 7 plus negative 28

00:04:05.630 --> 00:04:07.550
is negative 21.

00:04:07.550 --> 00:04:09.110
Well, this is a
nonsensical answer.

00:04:09.110 --> 00:04:11.270
0 can never be equal
to negative 21.

00:04:11.270 --> 00:04:13.710
No matter what x,
y, or z you pick,

00:04:13.710 --> 00:04:17.800
0 cannot be equal
to negative 21.

00:04:17.800 --> 00:04:21.355
And that's because these
second two equations

00:04:21.355 --> 00:04:22.730
right over here,
if you view them

00:04:22.730 --> 00:04:26.070
as planes in three dimensions,
these right over here

00:04:26.070 --> 00:04:27.050
do not intersect.

00:04:29.900 --> 00:04:31.740
If you visualize them
in three dimensions,

00:04:31.740 --> 00:04:34.130
they're actually
parallel planes.

00:04:34.130 --> 00:04:39.610
And since these last two
definitely do not intersect,

00:04:39.610 --> 00:04:44.360
we can say that this
system has no solutions.

00:04:44.360 --> 00:04:46.730
It doesn't matter if this
first equation intersects

00:04:46.730 --> 00:04:48.150
one or both of these.

00:04:48.150 --> 00:04:50.320
The fact that these
two don't intersect

00:04:50.320 --> 00:04:53.760
tells us that there's no unique
point x, y, z coordinate,

00:04:53.760 --> 00:04:57.170
a point in three dimensions that
satisfies all three of them.

00:04:57.170 --> 00:04:59.090
Because there's no
unique x, y, z that

00:04:59.090 --> 00:05:02.590
can satisfy these two, because
these are parallel planes.

00:05:02.590 --> 00:05:04.609
They do not intersect.


Ordered pair solutions to equations

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:04.180
Is 3 comma negative 4 a
solution to the equation

00:00:04.180 --> 00:00:07.047
5x plus 2y is equal to 7?

00:00:07.047 --> 00:00:08.630
So there's two ways
to think about it.

00:00:08.630 --> 00:00:11.230
One, you could just
substitute this x and y value

00:00:11.230 --> 00:00:13.780
into this equation to see if
it satisfies-- and then we'll

00:00:13.780 --> 00:00:15.670
do that way first--
and the other way

00:00:15.670 --> 00:00:18.140
is if you had a graph
of this equation,

00:00:18.140 --> 00:00:22.330
you could see if this point sits
on that graph, which would also

00:00:22.330 --> 00:00:24.872
mean that it is a
solution to this equation.

00:00:24.872 --> 00:00:26.080
So let's do it the first way.

00:00:26.080 --> 00:00:29.860
So we have 5x plus 2y is equal
to 7, so let's substitute.

00:00:29.860 --> 00:00:35.270
Instead of x, let
us put in 3 for x.

00:00:35.270 --> 00:00:44.850
So 5 times 3 plus 2 times
y-- so y is negative 4--

00:00:44.850 --> 00:00:50.262
plus 2 times negative 4
needs to be equal to 7.

00:00:50.262 --> 00:00:52.470
I'll put a question mark
here, because we're not sure

00:00:52.470 --> 00:00:53.610
yet if it does.

00:00:53.610 --> 00:01:05.200
So 5 times 3 is 15, and then 2
times negative 4 is negative 8.

00:01:05.200 --> 00:01:08.980
So the left hand side, it
simplifies to 15 minus 8,

00:01:08.980 --> 00:01:11.840
and this needs to be equal to 7.

00:01:11.840 --> 00:01:15.960
And of course, 15 minus 8 does
equal 7, so this all works out.

00:01:15.960 --> 00:01:18.975
This is a solution, so
we've answered the question.

00:01:18.975 --> 00:01:20.850
But I also want to show
you, this way we just

00:01:20.850 --> 00:01:22.220
did it by substitution.

00:01:22.220 --> 00:01:24.050
If we had the graph
of this equation,

00:01:24.050 --> 00:01:25.640
we could also do it graphically.

00:01:25.640 --> 00:01:28.560
So let's give ourselves
the graph of this equation,

00:01:28.560 --> 00:01:30.570
and I'll do that by
setting up a table.

00:01:30.570 --> 00:01:32.070
There's multiple
ways to graph this.

00:01:32.070 --> 00:01:34.153
You could put it in a
slope-intercept form and all

00:01:34.153 --> 00:01:37.180
of the rest, but I'll just set
up a table of x and y values.

00:01:40.030 --> 00:01:42.200
And I'll graph it, and
then given the graph,

00:01:42.200 --> 00:01:44.000
I want to see if this
actually sits on it.

00:01:44.000 --> 00:01:45.570
And obviously it will,
because we've already

00:01:45.570 --> 00:01:46.486
shown that this works.

00:01:46.486 --> 00:01:49.130
In fact, we could try
the point 3, negative 4,

00:01:49.130 --> 00:01:50.740
and that actually
is on the graph.

00:01:50.740 --> 00:01:53.190
We could do it on our table,
but I won't do that just yet.

00:01:53.190 --> 00:01:56.420
I'm just going to do this
to give ourselves a graph.

00:01:56.420 --> 00:02:00.350
So let's try it when
x is equal to 0.

00:02:00.350 --> 00:02:05.560
We have 5 times 0 plus
2 times y is equal to 7.

00:02:05.560 --> 00:02:08.280
So when x is equal
to 0, y is going

00:02:08.280 --> 00:02:11.770
to be-- so you're going to
have 0 plus 2y is equal to 7.

00:02:11.770 --> 00:02:17.190
y is going to be equal to 3.5.

00:02:17.190 --> 00:02:23.430
When x is equal to 1, you
have 5 plus 2y is equal to 7.

00:02:27.160 --> 00:02:28.950
If you subtract 5
from both sides,

00:02:28.950 --> 00:02:31.170
you get 2y is equal to 2.

00:02:31.170 --> 00:02:33.360
You get y is equal to 1.

00:02:33.360 --> 00:02:40.090
So when x is 1, y is 1,
and when x is-- well let's

00:02:40.090 --> 00:02:43.720
try-- well that's actually
enough for us to graph.

00:02:43.720 --> 00:02:46.430
We could keep doing more points.

00:02:46.430 --> 00:02:48.490
We could even put the
point 3, negative 4 there,

00:02:48.490 --> 00:02:50.920
but let's just try to graph
it in this very rough sense

00:02:50.920 --> 00:02:52.570
right here.

00:02:52.570 --> 00:03:00.150
So let me draw my x-axis,
and then this right over here

00:03:00.150 --> 00:03:03.380
is my y-axis.

00:03:03.380 --> 00:03:05.100
And let me draw
some points here.

00:03:05.100 --> 00:03:10.320
So let's say that this
is y is 1, 2, 3, 4.

00:03:10.320 --> 00:03:14.610
This is negative 1, negative
2, negative 3, negative 4.

00:03:14.610 --> 00:03:17.180
I could keep going
down in that direction.

00:03:17.180 --> 00:03:18.690
This is 1.

00:03:18.690 --> 00:03:23.360
Let me do it a little
bit-- 1, 2, 3, 4, and I

00:03:23.360 --> 00:03:26.416
could just keep going on and
on in the positive x direction.

00:03:26.416 --> 00:03:27.540
So let's plot these points.

00:03:27.540 --> 00:03:28.950
I have 0, 3.5.

00:03:28.950 --> 00:03:34.160
When x is 0, y is 1, 2, 3.5.

00:03:34.160 --> 00:03:38.500
When x is 1, y is 1.

00:03:38.500 --> 00:03:41.140
And so if we were to draw
this line-- I'll do it

00:03:41.140 --> 00:03:43.610
as a dotted line, just
so that I can make sure

00:03:43.610 --> 00:03:44.495
I connect the dots.

00:03:44.495 --> 00:03:47.030
I can do a better
job than that though.

00:03:47.030 --> 00:03:52.870
So it will look
something like that.

00:03:52.870 --> 00:03:55.510
And so if someone
gave you this line,

00:03:55.510 --> 00:03:58.080
you'd say oh, well it's 3,
negative 4 on this line,

00:03:58.080 --> 00:04:03.900
and let's assume that
we drew it really nicely

00:04:03.900 --> 00:04:05.060
and this was all to scale.

00:04:05.060 --> 00:04:09.530
Let me try one
last attempt at it.

00:04:09.530 --> 00:04:15.530
So it's going to look
something like that.

00:04:15.530 --> 00:04:17.730
And If someone asked
is 3, negative 4 on it,

00:04:17.730 --> 00:04:19.990
you could visually do
it, but it's always hard

00:04:19.990 --> 00:04:21.140
when you actually
don't substitute it,

00:04:21.140 --> 00:04:21.579
because you don't know.

00:04:21.579 --> 00:04:23.070
Maybe you're a little bit off.

00:04:23.070 --> 00:04:25.100
But if you look at
it over here, you

00:04:25.100 --> 00:04:27.710
say when x is equal
to 3, what is y?

00:04:27.710 --> 00:04:29.710
Well, you go down
here, and it looks

00:04:29.710 --> 00:04:32.800
like y is equal to negative 4.

00:04:32.800 --> 00:04:37.080
So this is a point
3 comma negative 4.

00:04:37.080 --> 00:04:39.980
Obviously, in general,
you don't want

00:04:39.980 --> 00:04:42.240
to just rely off of
inspecting graphs.

00:04:42.240 --> 00:04:45.781
Maybe this was 3,
negative 3.9999,

00:04:45.781 --> 00:04:47.780
and you just couldn't
tell looking at the graph.

00:04:47.780 --> 00:04:50.530
That's why you always want
to just the substitute

00:04:50.530 --> 00:04:52.840
and make sure that
it really does equal,

00:04:52.840 --> 00:04:56.210
that this equality really
does hold true at that point,

00:04:56.210 --> 00:04:57.460
not just looking at the graph.

00:04:57.460 --> 00:04:59.640
But it's important to
realize that the graph really

00:04:59.640 --> 00:05:03.170
is another representation
to all of the solutions

00:05:03.170 --> 00:05:05.310
of this equation.


Perpendicular Line Slope

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:03.490
Find the slope of a
line perpendicular

00:00:03.490 --> 00:00:07.510
to the line 3x plus
2y is equal to 6.

00:00:07.510 --> 00:00:10.560
So if I have one line-- so let
me draw some quick axes here.

00:00:10.560 --> 00:00:14.420
So that's my y-axis,
and that is my x-axis.

00:00:14.420 --> 00:00:18.920
And let's say I have a line
here that has a slope of m,

00:00:18.920 --> 00:00:24.360
so the slope over
here is equal to m.

00:00:24.360 --> 00:00:26.900
A line that is
perpendicular to this line

00:00:26.900 --> 00:00:29.000
would intersect
it at 90 degrees,

00:00:29.000 --> 00:00:31.900
so it would look
something like this.

00:00:31.900 --> 00:00:34.290
This would be a 90 degree angle.

00:00:34.290 --> 00:00:37.010
And the slope here-- and I
won't prove it in this video,

00:00:37.010 --> 00:00:39.480
we talk about it in other
videos-- the slope here

00:00:39.480 --> 00:00:42.430
is going to be equal to
the negative reciprocal

00:00:42.430 --> 00:00:44.520
of this line slope.

00:00:44.520 --> 00:00:49.470
So it's going to be the negative
reciprocal of that slope,

00:00:49.470 --> 00:00:53.610
or another way it's going to
be negative 1 over this slope,

00:00:53.610 --> 00:00:55.062
or the negative reciprocal.

00:00:55.062 --> 00:00:57.270
Now, with that out of the
way, if we could figure out

00:00:57.270 --> 00:00:59.070
the slope of this
line right over here,

00:00:59.070 --> 00:01:01.960
we would just have to take
the negative reciprocal,

00:01:01.960 --> 00:01:04.760
and we would have the slope
of the perpendicular line.

00:01:04.760 --> 00:01:07.125
So let's figure out
the slope of this line.

00:01:07.125 --> 00:01:08.000
So let me rewrite it.

00:01:08.000 --> 00:01:11.400
3x plus 2y is equal to 6.

00:01:11.400 --> 00:01:13.490
And the easiest thing
to do is to solve for y,

00:01:13.490 --> 00:01:16.670
so we get it into y
equals mx plus b form,

00:01:16.670 --> 00:01:18.930
and then the slope
will be pretty obvious.

00:01:18.930 --> 00:01:21.170
So let's get rid of the
x on the left-hand side,

00:01:21.170 --> 00:01:25.120
so let's subtract x-- 3x from
both sides, I should say.

00:01:25.120 --> 00:01:28.860
And then the left-hand side,
we're left with just a 2y.

00:01:28.860 --> 00:01:33.350
The right-hand side, we
have a negative 3x plus 6.

00:01:33.350 --> 00:01:36.046
Now we can divide
both sides by 2,

00:01:36.046 --> 00:01:38.420
so we have to divide all the
terms on the right-hand side

00:01:38.420 --> 00:01:39.610
by 2.

00:01:39.610 --> 00:01:44.490
We get y is equal to
negative 3/2x-- that's

00:01:44.490 --> 00:01:47.220
that part right
over there-- plus 3.

00:01:47.220 --> 00:01:49.310
So the slope of
this line right over

00:01:49.310 --> 00:01:52.050
here that they give us--
the slope of this line, now

00:01:52.050 --> 00:01:55.870
that we've put it in mx plus
b form, slope-intercept form,

00:01:55.870 --> 00:01:59.300
we know that the slope of
this line is negative 3/2.

00:01:59.300 --> 00:02:01.490
That is the slope.

00:02:01.490 --> 00:02:04.200
So the slope of a line that
is perpendicular to this

00:02:04.200 --> 00:02:06.850
will be the negative
reciprocal of this.

00:02:06.850 --> 00:02:10.210
So the negative of a
negative is a positive,

00:02:10.210 --> 00:02:11.810
so it's going to be a positive.

00:02:11.810 --> 00:02:14.690
And then the reciprocal
of 3/2 is going to be 2/3.

00:02:17.300 --> 00:02:20.630
So the negative
reciprocal of negative 3/2

00:02:20.630 --> 00:02:24.920
is positive 2/3, so we're done.

00:02:24.920 --> 00:02:28.510
That's the slope of a line that
is perpendicular to this one

00:02:28.510 --> 00:02:30.650
right over here.


Multiplying three numbers in scientific notation (example)

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.200
We're asked to
multiply 1.45 times 10

00:00:03.200 --> 00:00:06.900
to the eighth times 9.2 times
10 to the negative 12th times

00:00:06.900 --> 00:00:08.880
3.01 times 10 to
the negative fifth

00:00:08.880 --> 00:00:11.920
and express the product in
both decimal and scientific

00:00:11.920 --> 00:00:13.040
notation.

00:00:13.040 --> 00:00:19.760
So this is 1.45 times 10
to the eighth power times--

00:00:19.760 --> 00:00:22.374
and I could just write the
parentheses again like this,

00:00:22.374 --> 00:00:24.790
but I'm just going to write
it as another multiplication--

00:00:24.790 --> 00:00:29.970
times 9.2 times 10
to the negative 12th

00:00:29.970 --> 00:00:37.620
and then times 3.01 times
10 to the negative fifth.

00:00:37.620 --> 00:00:40.722
All this meant, when I wrote
these parentheses times next

00:00:40.722 --> 00:00:42.430
to each other, I'm
just going to multiply

00:00:42.430 --> 00:00:44.010
this expression
times this expression

00:00:44.010 --> 00:00:45.330
times this expression.

00:00:45.330 --> 00:00:47.590
And since everything is
involved multiplication,

00:00:47.590 --> 00:00:51.020
it actually doesn't matter
what order I multiply in.

00:00:51.020 --> 00:00:53.470
And so with that in mind,
I can swap the order here.

00:00:53.470 --> 00:00:57.390
This is going to be the
same thing as 1.45-- that's

00:00:57.390 --> 00:01:09.530
that right there-- times
9.2 times 3.01 times

00:01:09.530 --> 00:01:11.370
10 to the eighth--
let me do that

00:01:11.370 --> 00:01:18.490
in that purple color-- times
10 to the eighth times 10

00:01:18.490 --> 00:01:24.910
to the negative 12th power times
10 to the negative fifth power.

00:01:28.880 --> 00:01:30.750
And this is useful
because now I have

00:01:30.750 --> 00:01:32.850
all of my powers of
10 right over here.

00:01:32.850 --> 00:01:34.720
I could put parentheses
around that.

00:01:34.720 --> 00:01:38.460
And I have all my non-powers
of 10 right over there.

00:01:38.460 --> 00:01:39.730
And so I can simplify it.

00:01:39.730 --> 00:01:42.750
If I have the same base
10 right over here,

00:01:42.750 --> 00:01:44.620
so I can add the exponents.

00:01:44.620 --> 00:01:52.320
This is going to be 10 to
the 8 minus 12 minus 5 power.

00:01:54.960 --> 00:01:57.160
And then all of this
on the left-hand side--

00:01:57.160 --> 00:02:02.187
let me get a calculator
out-- I have 1.45.

00:02:02.187 --> 00:02:04.520
You could do it by hand, but
this is a little bit faster

00:02:04.520 --> 00:02:08.850
and less likely to make a
careless mistake-- times 9.2

00:02:08.850 --> 00:02:17.620
times 3.01, which
is equal to 40.1534.

00:02:17.620 --> 00:02:22.380
So this is equal to 40.1534.

00:02:22.380 --> 00:02:24.810
And of course, this is going
to be multiplied times 10

00:02:24.810 --> 00:02:25.950
to this thing.

00:02:25.950 --> 00:02:27.710
And so if we simplify
this exponent,

00:02:27.710 --> 00:02:33.630
you get 40.1534 times
10 to the 8 minus 12

00:02:33.630 --> 00:02:36.520
is negative 4, minus
5 is negative 9.

00:02:36.520 --> 00:02:39.180
10 to the negative 9 power.

00:02:39.180 --> 00:02:41.470
Now you might be tempted
to say that this is already

00:02:41.470 --> 00:02:44.150
in scientific notation because
I have some number here

00:02:44.150 --> 00:02:45.950
times some power of 10.

00:02:45.950 --> 00:02:49.820
But this is not quite
official scientific notation.

00:02:49.820 --> 00:02:51.530
And that's because
in order for it

00:02:51.530 --> 00:02:55.700
to be in scientific notation,
this number right over here

00:02:55.700 --> 00:03:01.060
has to be greater than or
equal to 1 and less than 10.

00:03:01.060 --> 00:03:03.420
And this is, obviously,
not less than 10.

00:03:03.420 --> 00:03:05.860
Essentially, for it to be
in scientific notation,

00:03:05.860 --> 00:03:08.495
you want a non-zero
digit right over here.

00:03:08.495 --> 00:03:10.120
And then you want
your decimal and then

00:03:10.120 --> 00:03:11.850
the rest of everything else.

00:03:11.850 --> 00:03:15.710
So here-- and you want
a non-zero single digit

00:03:15.710 --> 00:03:16.210
over here.

00:03:16.210 --> 00:03:18.930
Here we obviously
have two digits.

00:03:18.930 --> 00:03:22.910
This is larger than 10-- or this
is greater than or equal to 10.

00:03:22.910 --> 00:03:25.280
You want this thing
to be less than 10

00:03:25.280 --> 00:03:27.959
and greater than or equal to 1.

00:03:27.959 --> 00:03:30.000
So the best way to do that
is to write this thing

00:03:30.000 --> 00:03:32.260
right over here in
scientific notation.

00:03:32.260 --> 00:03:40.200
This is the same thing
as 4.01534 times 10.

00:03:40.200 --> 00:03:43.480
And one way to think about
it is to go from 40 to 4,

00:03:43.480 --> 00:03:46.990
we have to move this
decimal over to the left.

00:03:46.990 --> 00:03:49.400
Moving a decimal over to
the left to go from 40 to 4

00:03:49.400 --> 00:03:50.580
you're dividing by 10.

00:03:50.580 --> 00:03:53.470
So you have to multiply by
10 so it all equals out.

00:03:53.470 --> 00:03:55.539
Divide by 10 and
then multiply by 10.

00:03:55.539 --> 00:03:58.080
Or another way to write it, or
another way to think about it,

00:03:58.080 --> 00:04:03.820
is 4.0 and all this stuff times
10 is going to be 40.1534.

00:04:03.820 --> 00:04:06.260
And so you're going to have
4-- all of this times 10

00:04:06.260 --> 00:04:11.120
to the first power, that's
the same thing as 10-- times

00:04:11.120 --> 00:04:15.070
this thing-- times 10 to
the negative ninth power.

00:04:15.070 --> 00:04:17.640
And then once
again, powers of 10,

00:04:17.640 --> 00:04:19.890
so it's 10 to the first
times 10 to the negative 9

00:04:19.890 --> 00:04:24.800
is going to be 10 to the
negative eighth power.

00:04:24.800 --> 00:04:31.780
And we still have this 4.01534
times 10 to the negative 8.

00:04:31.780 --> 00:04:38.490
And now we have written
it in scientific notation.

00:04:38.490 --> 00:04:39.865
Now, they wanted
us to express it

00:04:39.865 --> 00:04:42.570
in both decimal and
scientific notation.

00:04:42.570 --> 00:04:45.150
And when they're asking us to
write it in decimal notation,

00:04:45.150 --> 00:04:48.810
they essentially want us to
multiply this out, expand this

00:04:48.810 --> 00:04:49.610
out.

00:04:49.610 --> 00:04:53.140
And so the way to think about
it-- write these digits out.

00:04:53.140 --> 00:04:58.020
So I have 4, 0, 1, 5, 3, 4.

00:04:58.020 --> 00:04:59.900
And if I'm just
looking at this number,

00:04:59.900 --> 00:05:02.420
I start with the
decimal right over here.

00:05:02.420 --> 00:05:08.360
Now, every time I divide by
10, or if I multiply by 10

00:05:08.360 --> 00:05:12.520
to the negative 1, I'm moving
this over to the left one spot.

00:05:12.520 --> 00:05:15.290
So 10 to the negative
1-- if I multiply by 10

00:05:15.290 --> 00:05:18.630
to the negative 1, that's the
same thing as dividing by 10.

00:05:18.630 --> 00:05:21.440
And so I'm moving the
decimal over to the left one.

00:05:21.440 --> 00:05:24.490
Here I'm multiplying by
10 to the negative 8.

00:05:24.490 --> 00:05:27.960
Or you could say I'm dividing
by 10 to the eighth power.

00:05:27.960 --> 00:05:30.901
So I'm going to want to move
the decimal to the left eight

00:05:30.901 --> 00:05:31.400
times.

00:05:41.749 --> 00:05:43.165
And one way to
remember it-- look,

00:05:43.165 --> 00:05:46.340
this is a very, very,
very, very small number.

00:05:46.340 --> 00:05:48.860
If I multiply this, I
should get a smaller number.

00:05:48.860 --> 00:05:51.360
So I should be moving
the decimal to the left.

00:05:51.360 --> 00:05:53.510
If this was a
positive 8, then this

00:05:53.510 --> 00:05:55.440
would be a very large number.

00:05:55.440 --> 00:05:57.510
And so if I multiply
by a large power of 10,

00:05:57.510 --> 00:05:59.510
I'm going to be moving
the decimal to the right.

00:05:59.510 --> 00:06:02.790
So this whole thing
should evaluate

00:06:02.790 --> 00:06:06.960
to being smaller than 4.01534.

00:06:06.960 --> 00:06:11.010
So I move the decimal
eight times to the left.

00:06:11.010 --> 00:06:13.770
I move it one time to the left
to get it right over here.

00:06:13.770 --> 00:06:16.890
And then the next seven times,
I'm just going to add 0's.

00:06:16.890 --> 00:06:22.710
So one, two, three, four,
five, six, seven 0's.

00:06:22.710 --> 00:06:25.510
And I'll put a 0 in front of
the decimal just to clarify it.

00:06:25.510 --> 00:06:29.020
So now I notice, if you include
this digit right over here,

00:06:29.020 --> 00:06:30.910
I have a total of eight digits.

00:06:33.930 --> 00:06:36.835
I have seven 0's, and
this digit gives us eight.

00:06:36.835 --> 00:06:41.494
So again, one, two, three,
four, five, six, seven, eight.

00:06:41.494 --> 00:06:42.910
The best way to
think about it is,

00:06:42.910 --> 00:06:44.630
I started with the
decimal right here.

00:06:44.630 --> 00:06:50.510
I moved once, twice, three,
four, five, six, seven,

00:06:50.510 --> 00:06:51.520
eight times.

00:06:51.520 --> 00:06:54.730
That's what multiplying times
10 to the negative 8 did for us.

00:06:54.730 --> 00:06:57.060
And I get this number
right over here.

00:06:57.060 --> 00:06:58.560
And when you see a
number like this,

00:06:58.560 --> 00:07:00.720
you start to appreciate
why we rewrite things

00:07:00.720 --> 00:07:02.920
in scientific notation.

00:07:02.920 --> 00:07:06.090
This is much easier to-- it
takes less space to write

00:07:06.090 --> 00:07:09.020
and you immediately know
roughly how big this number is.

00:07:09.020 --> 00:07:10.810
This is much harder to write.

00:07:10.810 --> 00:07:12.420
You might even
forget a 0 when you

00:07:12.420 --> 00:07:14.400
write it or you might add a 0.

00:07:14.400 --> 00:07:17.410
And now the person has to sit
and count the 0's to figure out

00:07:17.410 --> 00:07:20.950
essentially how large--or get
a rough sense of how large this

00:07:20.950 --> 00:07:21.450
thing is.

00:07:21.450 --> 00:07:24.660
It's one, two, three, four,
five, six, seven 0's, and you

00:07:24.660 --> 00:07:25.940
have this digit right here.

00:07:25.940 --> 00:07:27.440
That's what gets
us to that eight.

00:07:27.440 --> 00:07:31.250
But this is a much, much more
complicated-looking number

00:07:31.250 --> 00:07:34.076
than the one in
scientific notation.


Equation of a line from fractional slope and point

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:03.830
Write the equation of the
line that has a slope of 1/3

00:00:03.830 --> 00:00:08.490
and contains the point
negative 12, negative 14/3.

00:00:08.490 --> 00:00:11.080
So the equation of
a line we can write

00:00:11.080 --> 00:00:15.500
as y is equal to mx plus b.

00:00:15.500 --> 00:00:20.940
This is in slope-intercept
form, where m is our slope

00:00:20.940 --> 00:00:25.230
and b right over here
is our y-intercept.

00:00:25.230 --> 00:00:27.540
And they have actually
given us our slope.

00:00:27.540 --> 00:00:30.550
Our slope, they tell us, is 1/3.

00:00:30.550 --> 00:00:33.590
So we know that m
is going to be 1/3.

00:00:33.590 --> 00:00:39.720
So we know that y is
equal to 1/3 x plus b.

00:00:39.720 --> 00:00:41.870
And so now we just
have to solve for b.

00:00:41.870 --> 00:00:44.610
And the way we can solve for
b is that we know this line

00:00:44.610 --> 00:00:48.523
contains the point
negative 12, negative 14/3.

00:00:48.523 --> 00:00:54.410
So when x is equal
to negative 12,

00:00:54.410 --> 00:00:56.821
y is equal to negative 14/3.

00:01:02.114 --> 00:01:03.682
So let's write that down.

00:01:03.682 --> 00:01:10.700
Negative 14/3 is going
to be equal to 1/3--

00:01:10.700 --> 00:01:14.010
our slope-- times
our x value, which

00:01:14.010 --> 00:01:20.410
is negative 12-- times
negative 12-- and then plus b.

00:01:20.410 --> 00:01:23.120
So when x is
negative 12, we know

00:01:23.120 --> 00:01:25.450
that y is equal
to negative 14/3.

00:01:25.450 --> 00:01:27.300
And we also know that
the slope is 1/3.

00:01:27.300 --> 00:01:29.430
That's the m in mx plus b.

00:01:29.430 --> 00:01:31.440
So now we can just solve for b.

00:01:31.440 --> 00:01:35.770
The left-hand side-- I'll just
leave it as negative 14/3--

00:01:35.770 --> 00:01:40.480
is equal to 1/3
times negative 12.

00:01:40.480 --> 00:01:44.810
1/3 times negative
12 is negative 4,

00:01:44.810 --> 00:01:47.737
so this is negative 4, plus b.

00:01:47.737 --> 00:01:51.076
Let me write the plus
b in that same color

00:01:51.076 --> 00:01:52.433
that I had started it with.

00:01:52.433 --> 00:01:55.370
Plus b.

00:01:55.370 --> 00:01:58.940
It looks like a yellow color.

00:01:58.940 --> 00:02:02.590
And now, to isolate b
on the right-hand side,

00:02:02.590 --> 00:02:03.990
we can add 4 to both sides.

00:02:03.990 --> 00:02:05.531
I want to get rid
of this negative 4.

00:02:05.531 --> 00:02:07.160
So let's add 4 to both sides.

00:02:07.160 --> 00:02:12.920
So plus 4, plus 4-- on
the right-hand side,

00:02:12.920 --> 00:02:14.770
we're left with just a b.

00:02:14.770 --> 00:02:19.960
b is equal to-- on the
left-hand side, we have 4.

00:02:19.960 --> 00:02:21.380
4 I can rewrite.

00:02:21.380 --> 00:02:25.060
Because I'm going to have
to add 4 to negative 14/3.

00:02:25.060 --> 00:02:26.520
So I want a common denominator.

00:02:26.520 --> 00:02:28.990
4 is the same thing as 12/3.

00:02:28.990 --> 00:02:32.215
So 4 is 12/3, minus 14/3.

00:02:36.086 --> 00:02:37.210
So I just want to be clear.

00:02:37.210 --> 00:02:40.500
I just rewrote the 4 as 12/3.

00:02:40.500 --> 00:02:46.000
So 12/3 minus 14/3, or
negative 14/3 plus 12/3,

00:02:46.000 --> 00:02:50.550
is going to give us-- well, we
have the same denominator of 3.

00:02:50.550 --> 00:02:51.640
So we're going to have 3.

00:02:51.640 --> 00:02:54.590
12 minus 14 is negative 2.

00:02:54.590 --> 00:02:58.910
So we have negative
2/3 is equal to b.

00:02:58.910 --> 00:03:01.350
So the equation of the
line we figured out.

00:03:01.350 --> 00:03:02.320
They gave us the slope.

00:03:02.320 --> 00:03:04.500
And now we just figured
out the y-intercept

00:03:04.500 --> 00:03:06.861
using this information
right over here.

00:03:06.861 --> 00:03:08.360
So the equation of
our line is going

00:03:08.360 --> 00:03:15.050
to be y is equal to 1/3 x-- we
got the 1/3 from the problem--

00:03:15.050 --> 00:03:17.500
plus b. b is negative 2/3.

00:03:17.500 --> 00:03:21.000
So I could just write minus 2/3.

00:03:21.000 --> 00:03:23.920
And we are done.


Box-and-Whisker Plots

WEBVTT
Kind: captions
Language: en

00:00:00.640 --> 00:00:03.900
The owner of a restaurant
wants to find out more about

00:00:03.900 --> 00:00:06.020
where his patrons
are coming from.

00:00:06.020 --> 00:00:07.680
One day, he decided
to gather data

00:00:07.680 --> 00:00:09.780
about the distance
in miles that people

00:00:09.780 --> 00:00:11.780
commuted to get
to his restaurant.

00:00:11.780 --> 00:00:14.690
People reported the
following distances traveled.

00:00:14.690 --> 00:00:16.770
So this is our data
right over here.

00:00:16.770 --> 00:00:19.200
He wants to create a graph
that helps them understand

00:00:19.200 --> 00:00:22.840
the spread of the distances
and the median distance

00:00:22.840 --> 00:00:24.050
that people travel.

00:00:24.050 --> 00:00:26.640
What kind of a graph
should he create?

00:00:26.640 --> 00:00:29.600
And you can plot this data in
many different types of graphs,

00:00:29.600 --> 00:00:31.690
but they tend to depict
things in different ways.

00:00:31.690 --> 00:00:34.512
For example, a line graph
shows a trend over time.

00:00:34.512 --> 00:00:36.220
He's not interested
in a trend over time,

00:00:36.220 --> 00:00:37.678
so a line graph
doesn't make sense.

00:00:37.678 --> 00:00:40.807
Or a line graph could be a trend
of one variable with respect

00:00:40.807 --> 00:00:41.640
to another variable.

00:00:41.640 --> 00:00:42.973
It doesn't just have to be time.

00:00:42.973 --> 00:00:44.650
But he doesn't want
to see a trend here.

00:00:44.650 --> 00:00:46.233
A bar graph is good
when you're trying

00:00:46.233 --> 00:00:48.050
to bucket things, put
things into buckets

00:00:48.050 --> 00:00:50.440
and see how those
buckets are performing.

00:00:50.440 --> 00:00:52.820
Once again, that's not
exactly what he wants to see.

00:00:52.820 --> 00:00:55.490
A pie graph, if you
want to see kind

00:00:55.490 --> 00:00:57.420
of how things make up a whole.

00:00:57.420 --> 00:01:00.080
That's not what he wants
to see right over here.

00:01:00.080 --> 00:01:03.111
A stem and leaf plot does
help with distributions

00:01:03.111 --> 00:01:04.569
a little bit, but
it really doesn't

00:01:04.569 --> 00:01:08.870
tackle the median distance and
the spread really, really well.

00:01:08.870 --> 00:01:10.650
So the one that
does, and especially

00:01:10.650 --> 00:01:12.550
when people talk about
medians and spread,

00:01:12.550 --> 00:01:14.492
is a box and whiskers plot.

00:01:14.492 --> 00:01:16.200
And I'll show you how
to do it right now.

00:01:16.200 --> 00:01:18.180
Box and whiskers.

00:01:18.180 --> 00:01:20.630
And what a box and whisker
plot literally does

00:01:20.630 --> 00:01:22.620
is it shows the
spread of the data.

00:01:22.620 --> 00:01:24.079
It splits it into quartiles.

00:01:24.079 --> 00:01:25.620
And I'll talk about
that in a second.

00:01:25.620 --> 00:01:28.409
And it also tells you where the
median of the data actually is.

00:01:28.409 --> 00:01:30.200
And that's one of the
things that the owner

00:01:30.200 --> 00:01:31.802
of the restaurant cares about.

00:01:31.802 --> 00:01:33.510
So whenever you're
dealing with medians--

00:01:33.510 --> 00:01:36.720
and box and whisker plots deal
with medians-- the first thing

00:01:36.720 --> 00:01:38.430
you want to do is
order all the numbers

00:01:38.430 --> 00:01:40.555
because the median is really
the middle number when

00:01:40.555 --> 00:01:41.930
you order them all up.

00:01:41.930 --> 00:01:43.050
So let's order this.

00:01:43.050 --> 00:01:44.250
So let's write it in order.

00:01:44.250 --> 00:01:46.600
So first we have 1.

00:01:46.600 --> 00:01:48.030
So we get rid of that 1.

00:01:48.030 --> 00:01:51.290
Then we have one
2 right over here.

00:01:51.290 --> 00:01:54.490
And then we have another
2 right over there.

00:01:54.490 --> 00:01:55.890
That's all of our 2's.

00:01:55.890 --> 00:02:00.481
Then we have this 3, that
3, and then we have that 3.

00:02:00.481 --> 00:02:02.230
So those are two people
who have to travel

00:02:02.230 --> 00:02:03.762
3 miles to the restaurant.

00:02:03.762 --> 00:02:05.220
Then let's see, do
we have any 4's?

00:02:05.220 --> 00:02:07.400
We have one 4 right over there.

00:02:07.400 --> 00:02:11.740
And then we have another
4 right over there.

00:02:11.740 --> 00:02:14.920
Then, let's see,
do we have any 5's?

00:02:14.920 --> 00:02:17.532
Actually, I just realized
that I skipped one of the 1's.

00:02:17.532 --> 00:02:18.990
We have another 1
right over there.

00:02:18.990 --> 00:02:21.380
So let's write that
1 right over there.

00:02:21.380 --> 00:02:23.029
We actually had two 1's.

00:02:23.029 --> 00:02:23.820
And then let's see.

00:02:23.820 --> 00:02:25.140
Do we have any 5's?

00:02:25.140 --> 00:02:27.900
We do not have
any 5's over here.

00:02:27.900 --> 00:02:28.890
Do we have any 6's?

00:02:28.890 --> 00:02:31.630
We have one 6 right over here.

00:02:31.630 --> 00:02:34.290
And then we have no more 6's.

00:02:34.290 --> 00:02:35.880
Do we have any 7's?

00:02:35.880 --> 00:02:38.950
We have one 7 right over there.

00:02:38.950 --> 00:02:39.920
8's.

00:02:39.920 --> 00:02:42.601
We have an 8 right over there
and that's the only one.

00:02:42.601 --> 00:02:43.100
Let's see.

00:02:43.100 --> 00:02:44.510
Do we have any 9's?

00:02:44.510 --> 00:02:45.650
No 9's.

00:02:45.650 --> 00:02:46.510
Any 10's?

00:02:46.510 --> 00:02:48.870
Yes, we have a 10.

00:02:48.870 --> 00:02:49.900
Do we have an 11?

00:02:49.900 --> 00:02:51.040
Yep.

00:02:51.040 --> 00:02:52.250
11.

00:02:52.250 --> 00:02:53.160
Do we have any 12's?

00:02:53.160 --> 00:02:55.470
No 12's, no 13's.

00:02:55.470 --> 00:02:56.826
14?

00:02:56.826 --> 00:02:58.640
14.

00:02:58.640 --> 00:03:00.680
And we have a 15.

00:03:00.680 --> 00:03:02.100
People must like
this restaurant.

00:03:02.100 --> 00:03:03.308
They're traveling a good bit.

00:03:03.308 --> 00:03:07.730
Then we have a 20 and
then we have a 22.

00:03:07.730 --> 00:03:09.080
So I've ordered all the numbers.

00:03:09.080 --> 00:03:11.070
Let me make sure I
haven't skipped or gotten

00:03:11.070 --> 00:03:11.770
some duplicates.

00:03:11.770 --> 00:03:18.580
So 1, 2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16,

00:03:18.580 --> 00:03:22.630
17 people were surveyed, 17
patrons of the restaurant.

00:03:22.630 --> 00:03:29.330
And we have 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11, 12, 13, 14, 15,

00:03:29.330 --> 00:03:30.090
16, 17.

00:03:30.090 --> 00:03:30.590
All right.

00:03:30.590 --> 00:03:33.180
Seems like I've got all of
them and I've ordered them now.

00:03:33.180 --> 00:03:34.690
Now the median is
the middle number.

00:03:34.690 --> 00:03:38.310
We just said that we
had 17 of these numbers.

00:03:38.310 --> 00:03:40.150
So we want the
number-- and since it's

00:03:40.150 --> 00:03:43.242
an odd number of numbers,
the median actually

00:03:43.242 --> 00:03:44.450
will be one of these numbers.

00:03:44.450 --> 00:03:45.920
It's actually the middle number.

00:03:45.920 --> 00:03:47.980
It'll be the number
where eight are larger

00:03:47.980 --> 00:03:49.040
and eight are smaller.

00:03:49.040 --> 00:03:53.560
So 1, 2, 3, 4, 5, 6, 7, 8.

00:03:53.560 --> 00:03:55.544
So it looks like
this is our median.

00:03:55.544 --> 00:03:56.710
It will be the ninth number.

00:03:56.710 --> 00:04:01.710
And then you have 1, 2, 3, 4,
5, 6, 7, 8 that are larger.

00:04:01.710 --> 00:04:05.460
So eight are smaller than
6, eight are larger than 6.

00:04:05.460 --> 00:04:07.550
So 6 is the middle number.

00:04:07.550 --> 00:04:09.410
If we had an even
number of numbers here,

00:04:09.410 --> 00:04:11.629
if we had two middle
numbers, then we

00:04:11.629 --> 00:04:12.920
would take the average of them.

00:04:12.920 --> 00:04:15.686
When you have an odd
number of data points,

00:04:15.686 --> 00:04:17.269
then you can just
take the middle one.

00:04:17.269 --> 00:04:20.974
So this one right over
here is our median.

00:04:20.974 --> 00:04:23.140
And then when you do a box
plot, what you want to do

00:04:23.140 --> 00:04:26.040
is you want to find the
median of the set of numbers

00:04:26.040 --> 00:04:28.280
that are smaller
than the median.

00:04:28.280 --> 00:04:30.900
And you also want to find the
median of the set of numbers

00:04:30.900 --> 00:04:32.790
that are larger than the median.

00:04:32.790 --> 00:04:34.600
These are called
the first quartile

00:04:34.600 --> 00:04:35.600
and the second quartile.

00:04:35.600 --> 00:04:37.475
Because when you do
that, you split your data

00:04:37.475 --> 00:04:40.310
into four sections,
or quartiles.

00:04:40.310 --> 00:04:43.350
"Quar," you'd normally
associate with four.

00:04:43.350 --> 00:04:47.180
So let's look at the set
that's smaller than 6.

00:04:47.180 --> 00:04:50.850
So we have 1, 2, 3,
4, 5, 6, 7, 8 numbers.

00:04:50.850 --> 00:04:51.749
So it's an eight.

00:04:51.749 --> 00:04:53.290
If you have eight
data points, you're

00:04:53.290 --> 00:04:55.510
going to actually have
two middle numbers.

00:04:55.510 --> 00:04:59.010
So for example, you have
these two right over here

00:04:59.010 --> 00:05:02.830
are the two middle numbers--
three less, three more.

00:05:02.830 --> 00:05:04.960
You can't just pick the
2 because if you just

00:05:04.960 --> 00:05:07.364
picked the 2, you'd have
three less and four more.

00:05:07.364 --> 00:05:09.280
And you can't just pick
the 3 because then you

00:05:09.280 --> 00:05:11.390
would have three
more and four less.

00:05:11.390 --> 00:05:13.140
So these are our
middle two numbers.

00:05:13.140 --> 00:05:16.940
So the median of this group
right over here is 2.5.

00:05:16.940 --> 00:05:19.390
I averaged these
middle two numbers.

00:05:19.390 --> 00:05:22.550
And then let's do it over
here with this group.

00:05:22.550 --> 00:05:24.540
I'll do it in blue.

00:05:24.540 --> 00:05:26.470
So once again, we
have eight numbers.

00:05:26.470 --> 00:05:27.860
You're going to
have two middles.

00:05:27.860 --> 00:05:30.870
And so it's going to be the
third and the fourth number,

00:05:30.870 --> 00:05:33.150
which is 11 and 14.

00:05:33.150 --> 00:05:36.460
And if you average 11
and 14-- let's see.

00:05:36.460 --> 00:05:43.130
11 plus 14 is 25, divided
by 2 is 12.5, is 12.5.

00:05:43.130 --> 00:05:46.730
So this essentially divides
the data into four sections.

00:05:46.730 --> 00:05:49.700
You have everything up
to this first quartile.

00:05:49.700 --> 00:05:51.750
So you have this first
section right over here,

00:05:51.750 --> 00:05:55.540
this first median of the
lower half of the data is 2.5.

00:05:55.540 --> 00:05:59.010
Then you have everything between
2.5 and 6, then 6 to 12.5,

00:05:59.010 --> 00:06:01.390
and then everything
more than 12.5.

00:06:01.390 --> 00:06:03.050
And so what a box
and whisker plot

00:06:03.050 --> 00:06:05.370
is essentially a
graphical depiction

00:06:05.370 --> 00:06:06.680
of this right over here.

00:06:06.680 --> 00:06:08.030
So let's do that.

00:06:08.030 --> 00:06:10.010
I'm going to set
up a number line.

00:06:10.010 --> 00:06:13.990
So let me draw a number line
as straight as possible.

00:06:13.990 --> 00:06:16.730
So let's say that
is my number line.

00:06:16.730 --> 00:06:20.340
And let's say that this is 0.

00:06:20.340 --> 00:06:22.700
And let's say that this
is-- I want to make it.

00:06:22.700 --> 00:06:25.310
So let's say that this is 10.

00:06:25.310 --> 00:06:27.740
And then this would be 20.

00:06:27.740 --> 00:06:29.920
And then this would be 5.

00:06:29.920 --> 00:06:33.070
This would be-- let me do it a
little bit cleaner than that.

00:06:33.070 --> 00:06:34.630
This would be 15.

00:06:34.630 --> 00:06:37.270
25 would be right around there.

00:06:37.270 --> 00:06:39.530
So first thing on the
box and whiskers plot,

00:06:39.530 --> 00:06:41.360
you want to show the
entire range of data.

00:06:41.360 --> 00:06:44.060
So our smallest data
point starts at 1.

00:06:44.060 --> 00:06:45.300
So 1 is right about here.

00:06:45.300 --> 00:06:47.100
So if I do-- so this is 2.5.

00:06:47.100 --> 00:06:49.620
So 1 is right about here.

00:06:49.620 --> 00:06:51.700
And then our largest
data point is 22.

00:06:51.700 --> 00:06:54.480
22 would sit right about there.

00:06:54.480 --> 00:06:56.520
And I'll even label
it, although you often

00:06:56.520 --> 00:06:57.850
won't see it labeled like that.

00:06:57.850 --> 00:06:58.760
That's 1.

00:06:58.760 --> 00:07:01.330
And then that is 22.

00:07:01.330 --> 00:07:05.140
And then the middle half of
the data, we do with a box.

00:07:05.140 --> 00:07:07.190
So the middle half
is this quartile

00:07:07.190 --> 00:07:09.650
and this quartile
right over there.

00:07:09.650 --> 00:07:13.960
So the second quartile
starts at 2.5.

00:07:13.960 --> 00:07:17.410
So 2.5 is right there.

00:07:17.410 --> 00:07:21.100
So this is where we
start our box, 2.5.

00:07:21.100 --> 00:07:25.780
And then our third
quartile starts at 12.5.

00:07:25.780 --> 00:07:29.560
So 12.5 is right over there.

00:07:29.560 --> 00:07:32.320
12.5 is right over there.

00:07:32.320 --> 00:07:34.619
And then we can draw a box.

00:07:34.619 --> 00:07:36.160
I'll draw the box
in a neutral color.

00:07:36.160 --> 00:07:38.650
I'll do it actually
in that yellow color.

00:07:38.650 --> 00:07:40.160
So let me draw the box here.

00:07:40.160 --> 00:07:43.760
So the box shows where the
middle half of our data is.

00:07:43.760 --> 00:07:46.040
And now I can draw
these two arrows, these.

00:07:46.040 --> 00:07:47.224
So that's the box part.

00:07:47.224 --> 00:07:48.640
And then these two
arrows are what

00:07:48.640 --> 00:07:50.530
you would call the whiskers.

00:07:50.530 --> 00:07:52.350
And that shows where
all the other data is.

00:07:52.350 --> 00:07:54.320
So it's really showing
the spread of the data.

00:07:54.320 --> 00:07:57.480
And then the last thing we need
to show is the actual median.

00:07:57.480 --> 00:08:00.130
And the median-- I'll do it in
purple-- we already figured out

00:08:00.130 --> 00:08:01.250
is 6.

00:08:01.250 --> 00:08:03.980
So the median sits right
about-- see, this is 5.

00:08:03.980 --> 00:08:04.780
This would 7.5.

00:08:04.780 --> 00:08:09.070
6 would be right over there.

00:08:09.070 --> 00:08:10.790
So with this one
diagram, you've actually

00:08:10.790 --> 00:08:12.700
depicted all of this
information in terms

00:08:12.700 --> 00:08:14.390
of where is the median.

00:08:14.390 --> 00:08:16.130
The median is at 6.

00:08:16.130 --> 00:08:18.050
That is 6 right over there.

00:08:18.050 --> 00:08:19.910
Where is the middle
half of the data?

00:08:19.910 --> 00:08:29.600
Well, it's between 2.5 and
12.5, between 2.5 and 12.5.

00:08:29.600 --> 00:08:34.230
And all of the data, the entire
spread for all of the customers

00:08:34.230 --> 00:08:36.610
sits between-- and this is
what the whiskers do for us.

00:08:36.610 --> 00:08:40.706
It sits between 1 and
it sits between 22.

00:08:40.706 --> 00:08:43.039
And if you wanted to color
code it a little bit better--

00:08:43.039 --> 00:08:45.490
let me do that just
because it's fun.

00:08:45.490 --> 00:08:49.030
So this data right
over here-- and really,

00:08:49.030 --> 00:08:51.690
if you think about it, it's
including this data too.

00:08:51.690 --> 00:08:54.795
That's what this
whisker is depicting.

00:08:57.760 --> 00:09:00.900
This data right over-- Let me
do that in a different color.

00:09:00.900 --> 00:09:03.540
This data right
over here is kind

00:09:03.540 --> 00:09:06.077
of the first half of the box.

00:09:06.077 --> 00:09:07.160
That data right over here.

00:09:07.160 --> 00:09:09.490
Then you have your
median in magenta.

00:09:09.490 --> 00:09:15.030
Then this data right over here
is the second part of the box.

00:09:15.030 --> 00:09:19.090
So it's all of this
stuff right over here.

00:09:19.090 --> 00:09:21.370
And then finally, let
me pick a new color

00:09:21.370 --> 00:09:24.330
that I haven't used yet.

00:09:24.330 --> 00:09:28.980
This data is kind of
represented by this part,

00:09:28.980 --> 00:09:30.532
by this whisker right over here.

00:09:30.532 --> 00:09:32.240
Now there's one thing
I want to leave you

00:09:32.240 --> 00:09:35.700
with is that I used a method for
this box and whisker diagram.

00:09:35.700 --> 00:09:36.960
I found the median.

00:09:36.960 --> 00:09:40.870
I use a method where the two
halves, I got rid of the median

00:09:40.870 --> 00:09:42.460
and I found the
median of this part,

00:09:42.460 --> 00:09:43.959
and I found the
median of that part.

00:09:43.959 --> 00:09:45.380
And that's the
more typical method

00:09:45.380 --> 00:09:47.050
for box and whisker plots.

00:09:47.050 --> 00:09:49.010
Sometimes when you find
the median of the lower

00:09:49.010 --> 00:09:50.920
half and the upper
half, some people

00:09:50.920 --> 00:09:53.579
will include this median in both
sets when they calculate it.

00:09:53.579 --> 00:09:55.620
But I just want to let
you know that's out there.

00:09:55.620 --> 00:09:57.161
But the way that I
did it is actually

00:09:57.161 --> 00:09:58.200
the more mainstream way.

00:09:58.200 --> 00:10:00.570
That's the way most
calculators, I believe,

00:10:00.570 --> 00:10:03.868
actually will do a
box and whisker plot.