ck12.org normal distribution problems: Empirical rule

WEBVTT
Kind: captions
Language: en

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Let's do another problem from
the normal distribution section

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of ck12.org's AP
Statistics book.

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And I'm using this
because it's open source.

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It's actually quite a good book.

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The problems are, I think,
good practice for us.

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So let's see, number
three, number two.

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You can go to their
site, and I think

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you can download the book.

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Assume that the mean weight of
one-year-old girls in the US

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is normally distributed with
a mean of about 9.5 grams.

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That's got to be kilograms.

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I have a 10-month-old son,
and he weighs about 20 pounds,

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which is about 9 kilograms.

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9.5 grams is nothing.

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This would be if we were talking
about like mice or something.

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This has got to be kilograms.

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But anyway.

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It's about 9.5 kilograms
with a standard deviation

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of approximately 1.1 grams.

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So the mean is equal to 9.5
kilograms, I'm assuming,

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and the standard deviation
is equal to 1.1 grams.

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Without using a
calculator-- so that's

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an interesting clue--
estimate the percentage

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of one-year-old
girls in the US that

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meet the following condition.

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So when they say that--
"without a calculator estimate,"

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that's a big clue
or a big giveaway

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that we're supposed to
use the empirical rule,

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sometimes called the
68, 95, 99.7 rule.

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And if you remember, this
is the name of the rule.

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You've essentially
remembered the rule.

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What that tells us is if we
have a normal distribution--

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I'll do a bit of a
review here before we

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jump into this problem.

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If we have a normal
distribution--

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let me draw a
normal distribution.

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So it looks like that.

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That's my normal distribution.

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I didn't draw it perfectly,
but you get the idea.

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It should be symmetrical.

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This is our mean right there.

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That's our mean.

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If we go one standard
deviation above the mean,

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and one standard
deviation below the mean--

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so this is our mean plus
one standard deviation,

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this is our mean minus
one standard deviation--

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the probability of
finding a result,

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if we're dealing with a perfect
normal distribution that's

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between one standard deviation
below the mean and one

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standard deviation above the
mean, that would be this area.

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And it would be-- you
could guess-- 68%,

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68% chance you're
going to get something

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within one standard
deviation of the mean,

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either a standard deviation
below or above or anywhere

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in between.

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Now if we're talking about
two standard deviations

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around the mean--
so if we go down

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another standard deviation.

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So we go down another
standard deviation

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in that direction and
another standard deviation

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above the mean.

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And we were to ask
ourselves, what's

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the probability of finding
something within those two

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or within that range?

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Then it's, you
could guess it, 95%.

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And that includes this
middle area right here.

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So the 68% is a subset of 95%.

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And I think you know
where this is going.

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If we go three standard
deviations below the mean

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and above the mean, the
empirical rule, or the 68, 95,

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99.7 rule tells us
that there is a 99.7%

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chance of finding a result
in a normal distribution that

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is within three standard
deviations of the mean.

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So above three standard
deviations below the mean,

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and below three standard
deviations above the mean.

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That's what the
empirical rule tells us.

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Now, let's see if we can
apply it to this problem.

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So they gave us the mean
and the standard deviation.

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Let me draw that out.

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Let me draw my axis
first, as best as I can.

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That's my axis.

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Let me draw my bell curve.

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Let me draw the bell curve.

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That's about as
good of a bell curve

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as you can expect a
freehand drawer to do.

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And the mean here is-- and
this should be symmetric.

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This height should be the
same as that height, there.

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I think you get the idea.

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I'm not a computer.

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9.5 is the mean.

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I won't write the units.

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It's all in kilograms.

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One standard deviation
above the mean,

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we should add 1.1 to that.

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Because they told us the
standard deviation is 1.1.

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That's going to be 10.6.

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Let me just draw a
little dotted line there.

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Once standard deviation
below the mean,

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we're going to
subtract 1.1 from 9.5.

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And so that would be, what?

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8.4.

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If we go two standard
deviations above the mean,

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we would add another
standard deviation here.

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We went one standard deviation,
two standard deviations.

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That one goes to 11.7.

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And if we were to go
three standard deviations,

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we'd add 1.1 again.

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That would get us to 12.8.

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Doing that on the other
side-- one standard deviation

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below the mean is 8.4.

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Two standard deviations below
the mean, subtract 1.1 again,

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would be 7.3.

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And then three standard
deviations below the mean,

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it would be right there,
would be 6.2 kilograms.

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So that's our setup
for the problem.

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So what's the
probability that we

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would find a one-year-old
girl in the US

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that weighs less
than 8.4 kilograms?

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Or maybe I should say whose
mass is less than 8.4 kilograms.

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So if we look here, the
probability of finding a baby

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or a female baby that's
one-years-old with a mass

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or a weight of less
than 8.4 kilograms,

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that's this area right here.

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I said mass because kilograms
is actually a unit of mass.

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But most people use
it as weight, as well.

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So that's in that
area right there.

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So how can we
figure out that area

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under this normal distribution
using the empirical rule?

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Well, we know what this area is.

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We know what this area between
minus one standard deviation

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and plus one standard
deviation is.

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We know that that is 68%.

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And if that's 68%, then
that means in the parts that

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aren't in that middle
region, you have 32%.

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Because the area under the
entire normal distribution

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is 100, or 100%, or
1, depending on how

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you want to think about it.

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Because you can't have-- well,
all the possibilities combined

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can only add up to 1.

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You can't have more
than 100% there.

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So if you add up this leg
and this leg-- so this plus

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that leg is going
to be the remainder.

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So 100 minus 68, that's 32%.

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And 32% is if you add up this
left leg and this right leg

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over here.

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And this is a perfect
normal distribution.

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They told us it's
normally distributed.

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So it's going to be
perfectly symmetrical.

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So if this side and
that side add up to 32,

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but they're both
symmetrical-- meaning

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they have the exact
same area-- then

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this side right
here-- do it in pink.

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This side right
here-- it ended up

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looking more like
purple-- would be 16%.

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And this side right
here would be 16%.

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So your probability of
getting a result more than one

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standard deviation
above the mean--

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so that's this right-hand
side-- would be 16%.

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Or the probability
of having a result

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less than one standard deviation
below the mean-- that's

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this, right here, 16%.

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So they want to know the
probability of having a baby,

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at one-years-old, less
than 8.4 kilograms.

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Less than 8.4 kilograms
is this area right here,

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and that's 16%.

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So that's 16% for Part
A. Let's do Part B.

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Between 7.3 and 11.7
kilograms-- so between 7.3,

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that's right there.

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That's two standard
deviations below the mean.

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And 11.7-- it's two standard
deviations above the mean.

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So they're essentially
asking us what's

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the probability of getting
a result within two

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standard deviations of the mean.

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This was the mean, right here.

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This is two standard
deviations below.

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This is two standard
deviations above.

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Well, that's pretty
straightforward.

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The empirical rule
tells us-- between two

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standard deviations,
you have a 95% chance

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of getting bad results,
or a 95% chance

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of getting a result that is
within two standard deviations.

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So the empirical rule
just gives us that answer.

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And then finally, Part
C-- the probability

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of having a one-year-old US baby
girl more than 12.8 kilograms.

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So 12.8 kilograms is
three standard deviations

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above the mean.

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So we want to know the
probability of having a result

00:08:34.256 --> 00:08:36.380
more than three standard
deviations above the mean.

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So that is this area way out
there, that I drew in orange.

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Maybe I should do it
in a different color

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to really contrast it.

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So it's this long tail out
here, this little small area.

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So what is that probability?

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So let's turn back to
our empirical rule.

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Well, we know this area.

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We know the area between minus
three standard deviations

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and plus three
standard deviations.

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We know this.

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Since this is the last problem,
I can color the whole thing in.

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We know this area, right here--
between minus 3 and plus 3.

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That is 99.7%.

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The bulk of the results
fall under there-- I mean,

00:09:17.080 --> 00:09:18.402
almost all of them.

00:09:18.402 --> 00:09:20.360
So what do we have left
over for the two tails?

00:09:20.360 --> 00:09:21.610
Remember, there are two tails.

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This is one of them.

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And then you have
the results that

00:09:23.380 --> 00:09:25.180
are less than three
standard deviations

00:09:25.180 --> 00:09:27.620
below the mean, this
tail right there.

00:09:27.620 --> 00:09:32.610
So that tells us that this less
than three standard deviations

00:09:32.610 --> 00:09:35.870
below the mean and more than
three standard deviations

00:09:35.870 --> 00:09:39.290
above the mean combined
have to be the rest.

00:09:39.290 --> 00:09:43.300
Well, the rest--
it's only 0.3%.

00:09:43.300 --> 00:09:44.110
for the rest.

00:09:46.650 --> 00:09:48.280
And these two things
are symmetrical.

00:09:48.280 --> 00:09:49.760
They're going to be equal.

00:09:49.760 --> 00:09:54.750
So this right here it has to
be half of this, or 0.15%,

00:09:54.750 --> 00:09:59.270
and this, right here,
is going to be 0.15%.

00:09:59.270 --> 00:10:03.150
So the probability of
having a one-year-old baby

00:10:03.150 --> 00:10:06.740
girl in the US that is
more than 12.8 kilograms,

00:10:06.740 --> 00:10:08.640
if you assume a perfect
normal distribution,

00:10:08.640 --> 00:10:11.090
is the area under this
curve, the area that

00:10:11.090 --> 00:10:14.480
is more than three standard
deviations above the mean.

00:10:14.480 --> 00:10:21.830
And that is 0.15%.

00:10:21.830 --> 00:10:24.720
Anyway, hope you
found that useful.


ck12.org normal distribution problems: z-score

WEBVTT
Kind: captions
Language: en

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Here's the second problem
from CK12.org's AP

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statistics FlexBook.

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It's an open source
textbook, essentially.

00:00:10.570 --> 00:00:13.310
I'm using it essentially to
get some practice on some

00:00:13.310 --> 00:00:14.330
statistics problems.

00:00:14.330 --> 00:00:15.770
So here, number 2.

00:00:15.770 --> 00:00:19.070
The grades on a statistics
midterm for a high school are

00:00:19.070 --> 00:00:22.420
normally distributed with a
mean of 81 and a standard

00:00:22.420 --> 00:00:24.370
deviation of 6.3.

00:00:24.370 --> 00:00:25.220
All right.

00:00:25.220 --> 00:00:28.520
Calculate the z-scores for each
of the following exam grades.

00:00:28.520 --> 00:00:32.480
Draw and label a sketch
for each example.

00:00:32.480 --> 00:00:34.300
We can probably do it all
on the same example.

00:00:34.300 --> 00:00:36.310
But the first thing we'd
have to do is just remember

00:00:36.310 --> 00:00:37.750
what is a z-score.

00:00:37.750 --> 00:00:39.110
What is a z-score?

00:00:39.110 --> 00:00:43.120
A z-score is literally just
measuring how many standard

00:00:43.120 --> 00:00:50.265
deviations away from the mean?

00:00:55.940 --> 00:00:56.670
Just like that.

00:00:56.670 --> 00:00:58.260
So we literally just have to
calculate how many standard

00:00:58.260 --> 00:01:01.040
deviations each of these guys
are from the mean, and

00:01:01.040 --> 00:01:03.270
that's their z-scores.

00:01:03.270 --> 00:01:07.280
So let me do part a.

00:01:07.280 --> 00:01:09.210
So we have 65.

00:01:09.210 --> 00:01:12.570
So first we can just figure out
how far is 65 from the mean.

00:01:12.570 --> 00:01:14.980
Let me just draw one chart
here that we can use

00:01:14.980 --> 00:01:15.900
the entire time.

00:01:15.900 --> 00:01:19.135
So it's just our distribution.

00:01:19.135 --> 00:01:20.030
Let's see.

00:01:20.030 --> 00:01:21.050
We have a mean of 81.

00:01:24.050 --> 00:01:26.030
That's our mean.

00:01:26.030 --> 00:01:27.940
And then a standard
deviation of 6.3.

00:01:27.940 --> 00:01:30.550
So our distribution, they're
telling us that it's

00:01:30.550 --> 00:01:31.630
normally distributed.

00:01:31.630 --> 00:01:34.670
So I can draw a nice
bell curve here.

00:01:34.670 --> 00:01:37.630
They're saying it's normally
distributed, so that's as

00:01:37.630 --> 00:01:40.500
good of a bell curve as
I'm capable of drawing.

00:01:40.500 --> 00:01:43.060
This is the mean
right there at 81.

00:01:43.060 --> 00:01:45.700
And the standard
deviation is 6.3.

00:01:45.700 --> 00:01:50.140
So one standard deviation above
and below is going to be

00:01:50.140 --> 00:01:52.950
6.3 away from that mean.

00:01:52.950 --> 00:01:57.400
So if we go 6.3 in the positive
direction, that value right

00:01:57.400 --> 00:02:01.620
there is going to be 87.3.

00:02:01.620 --> 00:02:05.680
If we go 6.3 in the
negative direction,

00:02:05.680 --> 00:02:07.140
where does that get us?

00:02:07.140 --> 00:02:11.970
What, 74.7?

00:02:11.970 --> 00:02:15.190
Right, if we add 6, it'll
get us to 80.7, and then

00:02:15.190 --> 00:02:16.680
0.3 will get us to 81.

00:02:16.680 --> 00:02:18.920
So that's one standard
deviation below and above the

00:02:18.920 --> 00:02:21.560
mean, and then you'd add
another 6.3 to go 2 standard

00:02:21.560 --> 00:02:23.590
deviations, so on and so forth.

00:02:23.590 --> 00:02:26.890
So that's a drawing of
the distribution itself.

00:02:26.890 --> 00:02:28.640
So let's figure out
the z-scores for each

00:02:28.640 --> 00:02:34.110
of these grades.

00:02:34.110 --> 00:02:35.400
65 is how far?

00:02:35.400 --> 00:02:38.760
65 is maybe going to
be here someplace.

00:02:38.760 --> 00:02:40.430
So we first want to say,
well how far is it

00:02:40.430 --> 00:02:42.680
just from our mean?

00:02:42.680 --> 00:02:46.140
So the distance is, you just
want to positive number here.

00:02:46.140 --> 00:02:48.050
Well actually, you want
a negative number.

00:02:48.050 --> 00:02:50.560
Because you want your z-score
to be positive or negative.

00:02:50.560 --> 00:02:54.170
Negative would mean to the left
of the mean and positive would

00:02:54.170 --> 00:02:56.090
mean to the right of the mean.

00:02:56.090 --> 00:03:00.690
So we say 65 minus 81.

00:03:00.690 --> 00:03:03.000
So that's literally
how far away we are.

00:03:03.000 --> 00:03:05.750
But we want that in terms
of standard deviations.

00:03:05.750 --> 00:03:10.120
So we divide that by the length
or the magnitude of our

00:03:10.120 --> 00:03:11.290
standard deviation.

00:03:11.290 --> 00:03:14.290
So 65 minus 81.

00:03:14.290 --> 00:03:17.260
Let's see, 81 minus 65 is what?

00:03:17.260 --> 00:03:20.480
It is 5 plus 11.

00:03:20.480 --> 00:03:21.230
It's 16.

00:03:21.230 --> 00:03:26.065
So this is going to be
minus 16 over 6.3.

00:03:26.065 --> 00:03:28.900
We'll take our calculator out.

00:03:28.900 --> 00:03:37.060
And let's see, if we have
minus 16 divided by 6.3,

00:03:37.060 --> 00:03:40.980
you get minus 2 point--
oh, it's like 54.

00:03:40.980 --> 00:03:45.940
Approximately equal
to minus 2.54.

00:03:45.940 --> 00:03:50.010
That's the z-score
for a grade of 65.

00:03:50.010 --> 00:03:50.840
Pretty straightforward.

00:03:50.840 --> 00:03:52.120
Let's do a couple more.

00:03:52.120 --> 00:03:53.340
Let's do all of them.

00:03:53.340 --> 00:03:54.940
83.

00:03:54.940 --> 00:03:56.520
So how is it away
from the mean?

00:03:56.520 --> 00:03:59.040
Well, it's 83 minus 81.

00:03:59.040 --> 00:04:01.220
It's two grades above the mean.

00:04:01.220 --> 00:04:03.320
But we want it in terms
of standard deviations.

00:04:03.320 --> 00:04:04.960
How many standard deviations.

00:04:04.960 --> 00:04:05.960
So this was part A.

00:04:05.960 --> 00:04:08.310
A was right here.

00:04:08.310 --> 00:04:11.300
We were 2.5 standard
deviations below the mean.

00:04:11.300 --> 00:04:12.120
So this is part A.

00:04:12.120 --> 00:04:14.650
1, 2, and then 0.5.

00:04:14.650 --> 00:04:17.830
So this was A right there, 65.

00:04:17.830 --> 00:04:21.770
And then part B, 83, 83 is
going to be right here.

00:04:21.770 --> 00:04:24.560
A little bit higher,
but right here.

00:04:24.560 --> 00:04:31.610
And the z-score here, 83 minus
81 divided by 6.3 will get us--

00:04:31.610 --> 00:04:34.340
let's see, clear
the calculator.

00:04:34.340 --> 00:04:39.420
So we have 83 minus 81
is 2 divided by 6.3.

00:04:39.420 --> 00:04:44.280
It's 0.32, roughly.

00:04:44.280 --> 00:04:46.770
So here we get 0.32.

00:04:46.770 --> 00:04:52.520
So 83 is 0.32 standard
deviations above the mean.

00:04:52.520 --> 00:04:54.860
And so it would be roughly 1/3
third of the standard deviation

00:04:54.860 --> 00:04:55.685
along the way, right?

00:04:55.685 --> 00:04:57.710
Because this as one whole
standard deviation.

00:04:57.710 --> 00:05:02.260
So we're 0.3 of a standard
deviation above the mean.

00:05:02.260 --> 00:05:03.900
Choice number C.

00:05:03.900 --> 00:05:06.910
Or not choice, part C, I
guess I should call it.

00:05:06.910 --> 00:05:08.350
93.

00:05:08.350 --> 00:05:09.750
Well, we do the same exercise.

00:05:09.750 --> 00:05:13.670
93 is how much above the mean?

00:05:13.670 --> 00:05:16.735
Well, it's 93 minus 81 is 12.

00:05:16.735 --> 00:05:19.170
But we want it in terms
of standard deviations.

00:05:19.170 --> 00:05:22.090
So 12 is how many standard
deviations above the mean?

00:05:22.090 --> 00:05:24.230
Well, it's going
to be almost 2.

00:05:24.230 --> 00:05:25.680
Let's take the calculator out.

00:05:25.680 --> 00:05:29.680
So we get 12 divided by 6.3.

00:05:29.680 --> 00:05:31.790
It's 1.9 standard deviations.

00:05:31.790 --> 00:05:35.510
Its z-score is 1.9.

00:05:35.510 --> 00:05:39.020
Which means it's 1.9 standard
deviations above the mean.

00:05:39.020 --> 00:05:42.660
So the mean is 81, we go one
whole standard deviation, and

00:05:42.660 --> 00:05:45.900
then 0.9 standard deviations,
and that's where a score of

00:05:45.900 --> 00:05:47.840
93 would lie, right there.

00:05:47.840 --> 00:05:49.450
Its z-score is 1.9.

00:05:49.450 --> 00:05:51.450
And all that means is
1.9 standard deviations

00:05:51.450 --> 00:05:52.170
above the mean.

00:05:52.170 --> 00:05:53.800
Let's do the last one.

00:05:53.800 --> 00:05:55.990
I'll do it in magenta.

00:05:55.990 --> 00:05:57.670
D, part D.

00:05:57.670 --> 00:06:00.700
A score of 100.

00:06:00.700 --> 00:06:02.470
We don't even need
the problem anymore.

00:06:02.470 --> 00:06:04.220
A score of 100.

00:06:04.220 --> 00:06:04.920
Well, same thing.

00:06:04.920 --> 00:06:08.190
We figure out how far is 100
above the mean-- remember, the

00:06:08.190 --> 00:06:12.200
mean was 81-- and we divide
that by the length or the size

00:06:12.200 --> 00:06:14.260
or the magnitude of our
standard deviation.

00:06:14.260 --> 00:06:18.600
So 100 minus 81 is
equal to 19 over 6.3.

00:06:18.600 --> 00:06:23.140
So it's going to be a little
over 3 standard deviations.

00:06:23.140 --> 00:06:25.200
And in the next problem we'll
see what does that imply in

00:06:25.200 --> 00:06:27.940
terms of the probability of
that actually occurring.

00:06:27.940 --> 00:06:30.750
But if we just want to figure
out the z-score, 19 divided

00:06:30.750 --> 00:06:35.670
by 6.3 is equal to 3.01.

00:06:35.670 --> 00:06:37.290
So it's very close.

00:06:37.290 --> 00:06:39.690
3.02, really, if
I were to round.

00:06:39.690 --> 00:06:43.040
So it's very close to 3.02.

00:06:43.040 --> 00:06:48.500
Its z-score is 3.02, or a grade
of 100 is 3.02 standard

00:06:48.500 --> 00:06:49.830
deviations above the mean.

00:06:49.830 --> 00:06:54.800
So remember, this was the
mean right here at 81.

00:06:54.800 --> 00:06:57.870
We go 1 standard deviation
above the mean, 2 standard

00:06:57.870 --> 00:07:00.030
deviations above the mean, the
third standard deviation above

00:07:00.030 --> 00:07:01.030
the mean is right there.

00:07:01.030 --> 00:07:02.980
So we're sitting right
there on our chart.

00:07:02.980 --> 00:07:05.980
A little bit above that, 3.02
standard deviations above

00:07:05.980 --> 00:07:07.860
the mean, that's where
a score of 100 will be.

00:07:07.860 --> 00:07:09.560
And you can see the
probability, the height of

00:07:09.560 --> 00:07:11.390
this-- that's what the chart
tells us-- it's actually a

00:07:11.390 --> 00:07:12.500
very low probability.

00:07:12.500 --> 00:07:14.830
Actually, not just a very
low probability of getting

00:07:14.830 --> 00:07:16.000
something higher than that.

00:07:16.000 --> 00:07:19.460
Because as we learned before,
in a probably density function,

00:07:19.460 --> 00:07:22.730
if this is a continuous, not a
discreet, the probability of

00:07:22.730 --> 00:07:25.650
getting exactly that is 0,
if this wasn't discrete.

00:07:25.650 --> 00:07:28.050
But since this is scores
on a test, we know that

00:07:28.050 --> 00:07:30.010
it's actually a discrete
probability function.

00:07:30.010 --> 00:07:32.850
But the probability is low of
getting higher than that,

00:07:32.850 --> 00:07:34.760
because you can see where
we sit on the bell curve.

00:07:34.760 --> 00:07:38.300
Well anyway, hopefully this at
least clarified how to solve

00:07:38.300 --> 00:07:41.635
for z-scores, which is pretty
straightforward mathematically.

00:07:41.635 --> 00:07:44.440
And in the next video,
we'll interpret z-scores

00:07:44.440 --> 00:07:47.190
and probabilities
a little bit more.


ck12.org normal distribution problems: Qualitative sense of normal distributions

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.010
You can never have too
much practice dealing

00:00:03.010 --> 00:00:05.160
with the normal distribution,
because it's really

00:00:05.160 --> 00:00:07.470
one of those super
important building

00:00:07.470 --> 00:00:09.320
blocks for the
rest of statistics

00:00:09.320 --> 00:00:11.072
and really a lot of your life.

00:00:11.072 --> 00:00:13.280
So what I've done is I've
taken some sample problems.

00:00:13.280 --> 00:00:18.800
This is from ck12.org
Open Source flex book,

00:00:18.800 --> 00:00:20.344
their AP Statistics flex book.

00:00:20.344 --> 00:00:22.760
And I've taken the problems
from their normal distribution

00:00:22.760 --> 00:00:23.260
chapter.

00:00:23.260 --> 00:00:25.550
So you could go to their
site and actually look up

00:00:25.550 --> 00:00:27.410
these same problems.

00:00:27.410 --> 00:00:31.010
So this first problem, which
of the following data sets

00:00:31.010 --> 00:00:33.830
is most likely to be
normally distributed?

00:00:33.830 --> 00:00:35.490
For the other
choices, explain why

00:00:35.490 --> 00:00:38.232
you believe they would not
follow a normal distribution.

00:00:38.232 --> 00:00:38.940
So it's a choice.

00:00:38.940 --> 00:00:41.450
So this is my beliefs
come into play.

00:00:41.450 --> 00:00:43.550
So this is unusual
in the math context.

00:00:43.550 --> 00:00:44.970
It's more of a, what do I think?

00:00:44.970 --> 00:00:46.430
It's kind of an essay question.

00:00:46.430 --> 00:00:47.840
So let's see what
they have here.

00:00:47.840 --> 00:00:49.450
A, the hand span.

00:00:49.450 --> 00:00:51.280
Measured from the
tip of the thumb

00:00:51.280 --> 00:00:53.376
to the tip of the
extended fifth finger.

00:00:53.376 --> 00:00:54.750
So I think they're
talking about,

00:00:54.750 --> 00:00:57.160
let me see if I can draw a hand.

00:00:57.160 --> 00:00:59.407
So that's the index finger.

00:00:59.407 --> 00:01:00.865
And then you got
the middle finger.

00:01:00.865 --> 00:01:02.510
And then you got
your ring finger.

00:01:02.510 --> 00:01:04.190
And you got your pinky.

00:01:04.190 --> 00:01:06.080
And the hand will look
something like that.

00:01:06.080 --> 00:01:07.913
I think they're talking
about this distance.

00:01:07.913 --> 00:01:09.620
From the tip of the
thumb to the tip

00:01:09.620 --> 00:01:11.470
of the extended
fifth finger, which

00:01:11.470 --> 00:01:14.220
is a fancy way of saying
the pinky, I think.

00:01:14.220 --> 00:01:16.220
They're talking about
that distance right there.

00:01:16.220 --> 00:01:18.060
And they're saying, if
I were to measure it

00:01:18.060 --> 00:01:19.910
of a random sample of
high school seniors,

00:01:19.910 --> 00:01:21.240
what would it look like?

00:01:21.240 --> 00:01:23.170
Well, you know how far this is.

00:01:23.170 --> 00:01:25.250
This is a combination
of genetics

00:01:25.250 --> 00:01:28.610
and environmental factors,
maybe how much milk you drank

00:01:28.610 --> 00:01:31.010
or how much you hung from
your pinky from a bar

00:01:31.010 --> 00:01:32.540
while you were growing up.

00:01:32.540 --> 00:01:34.720
So I would think
that it is a sum

00:01:34.720 --> 00:01:36.980
of a huge number of
random processes.

00:01:36.980 --> 00:01:39.900
So I would guess that it is
roughly, normally distributed.

00:01:39.900 --> 00:01:43.240
You know, if I look at my own
hand, and my hand I don't think

00:01:43.240 --> 00:01:45.920
has grown much since I
was a high school senior.

00:01:45.920 --> 00:01:49.920
It looks like roughly
nine inches or so.

00:01:49.920 --> 00:01:50.570
I play guitar.

00:01:50.570 --> 00:01:53.090
Maybe that helped
me stretch my hand.

00:01:53.090 --> 00:01:55.960
But it's really
an essay question,

00:01:55.960 --> 00:01:57.477
so I just have to
say what I feel.

00:01:57.477 --> 00:01:59.060
So I would guess
that the distribution

00:01:59.060 --> 00:02:00.160
would look something like this.

00:02:00.160 --> 00:02:00.590
I don't know.

00:02:00.590 --> 00:02:01.464
I've never done this.

00:02:01.464 --> 00:02:06.450
But maybe it has a mean of
eight inches or nine inches

00:02:06.450 --> 00:02:09.530
and is distributed
something like this.

00:02:09.530 --> 00:02:11.820
It's distributed
something like that.

00:02:11.820 --> 00:02:14.590
So maybe it probably does look
like a normal distribution.

00:02:14.590 --> 00:02:16.600
But probably won't be
a perfect, in fact, I

00:02:16.600 --> 00:02:19.600
can guarantee you it won't be
a perfect normal distribution.

00:02:19.600 --> 00:02:25.850
Because one, no one can have
negative length of that span.

00:02:25.850 --> 00:02:27.490
This distance could
never be negative.

00:02:27.490 --> 00:02:30.030
So they're going
to have, I guess

00:02:30.030 --> 00:02:33.164
you could have no hand so that
would maybe be counted as 0.

00:02:33.164 --> 00:02:35.580
But the distribution wouldn't
go into the negative domain.

00:02:35.580 --> 00:02:39.464
So it wouldn't be a
perfect normal distribution

00:02:39.464 --> 00:02:40.380
on the left hand side.

00:02:40.380 --> 00:02:42.262
It would really
just end here at 0.

00:02:42.262 --> 00:02:43.845
And even on the right
hand side, there

00:02:43.845 --> 00:02:46.540
is some physically
impossible hand lengths.

00:02:46.540 --> 00:02:49.690
No one can have a
hand that's larger

00:02:49.690 --> 00:02:52.550
than the height
of the atmosphere

00:02:52.550 --> 00:02:54.220
or an astronomical unit.

00:02:54.220 --> 00:02:56.540
You would you start
touching the sun.

00:02:56.540 --> 00:02:59.020
There's some point, which
is physically impossible

00:02:59.020 --> 00:02:59.760
to get to.

00:02:59.760 --> 00:03:01.800
And in a true
normal distribution,

00:03:01.800 --> 00:03:03.310
if I were to flip
a bunch of coins

00:03:03.310 --> 00:03:05.900
there's some very,
very small probability

00:03:05.900 --> 00:03:08.400
that I could get a
million heads in a row.

00:03:08.400 --> 00:03:09.150
It's almost 0.

00:03:09.150 --> 00:03:10.390
But there's some probability.

00:03:10.390 --> 00:03:14.210
But in the case of hand span,
there's no way out here,

00:03:14.210 --> 00:03:16.650
you know, the probability
of human being

00:03:16.650 --> 00:03:18.860
who happens to be a high
school senior, having

00:03:18.860 --> 00:03:24.597
a one-mile length
hand span, that's 0.

00:03:24.597 --> 00:03:26.805
So it's not going to be a
perfect normal distribution

00:03:26.805 --> 00:03:29.590
of the outliers or as we
get further and further away

00:03:29.590 --> 00:03:30.480
from the mean.

00:03:30.480 --> 00:03:33.610
But I think it'll be a pretty
good, in our everyday world as

00:03:33.610 --> 00:03:35.360
good as we're going
to get, approximation.

00:03:35.360 --> 00:03:36.943
The normal distribution
is going to be

00:03:36.943 --> 00:03:39.745
a pretty good approximation for
the distribution that we see.

00:03:39.745 --> 00:03:42.380
And I guess one thing that
you know-- It's funny.

00:03:42.380 --> 00:03:44.940
This is high school
seniors, when I did this,

00:03:44.940 --> 00:03:48.210
it was kind of from my
point of view as a guy.

00:03:48.210 --> 00:03:50.650
And I would argue that high
school seniors, guys, probably

00:03:50.650 --> 00:03:52.070
have larger hands than women.

00:03:52.070 --> 00:03:54.974
So it's possible you actually
have a bimodal distribution.

00:03:54.974 --> 00:03:56.390
So instead of
having it like this,

00:03:56.390 --> 00:03:59.250
it's possible that the
distribution looks like this.

00:03:59.250 --> 00:04:02.610
That you have one peak for
guys, maybe at eight inches.

00:04:02.610 --> 00:04:07.330
And then maybe a peak for
women at seven inches.

00:04:07.330 --> 00:04:10.060
And then the distribution
falls off like that.

00:04:10.060 --> 00:04:11.820
So it's also possible
it could be bimodal.

00:04:11.820 --> 00:04:14.170
But in general, a
normal distribution

00:04:14.170 --> 00:04:16.459
is going to be a pretty
good approximation

00:04:16.459 --> 00:04:19.300
for part A of this problem.

00:04:19.300 --> 00:04:22.740
Let's see what part B, what
they're asking us to describe.

00:04:22.740 --> 00:04:25.150
The annual salaries
of all employees

00:04:25.150 --> 00:04:27.650
of a large shipping company.

00:04:27.650 --> 00:04:30.430
So if we're talking
about annual salaries,

00:04:30.430 --> 00:04:32.380
we have minimum
wage laws whatnot.

00:04:32.380 --> 00:04:34.338
so I would guess that
any corporation, if we're

00:04:34.338 --> 00:04:36.100
talking about full
time workers at least,

00:04:36.100 --> 00:04:39.277
there's going to be some
minimum salary that people have.

00:04:39.277 --> 00:04:41.110
So I would say, and
probably a lot of people

00:04:41.110 --> 00:04:42.901
will have that minimum
salary because it'll

00:04:42.901 --> 00:04:45.660
be probably the most
labor intensive jobs,

00:04:45.660 --> 00:04:47.850
you probably have
most people down there

00:04:47.850 --> 00:04:49.400
at the low end of the scale.

00:04:49.400 --> 00:04:52.460
And then you have your
different middle level managers

00:04:52.460 --> 00:04:53.520
and whatnot.

00:04:53.520 --> 00:04:56.040
And then you probably
have this big gap.

00:04:56.040 --> 00:04:59.700
And then you probably have
your true executives, maybe

00:04:59.700 --> 00:05:02.120
your CEO or whatnot.

00:05:02.120 --> 00:05:08.130
If this mean, right here,
is maybe $40,000 a year

00:05:08.130 --> 00:05:11.240
and this is probably
$80,000 where

00:05:11.240 --> 00:05:13.020
some of the mid-level
managers lie.

00:05:13.020 --> 00:05:15.397
But this out here, this
will probably be-- Actually

00:05:15.397 --> 00:05:17.980
if you were to draw it real, the
way I've scaled it right now,

00:05:17.980 --> 00:05:21.240
this would be about
$200,00, which is actually

00:05:21.240 --> 00:05:23.882
a reasonable salary for a CEO.

00:05:23.882 --> 00:05:25.590
But the reality is,
is that this actually

00:05:25.590 --> 00:05:27.660
might get pushed
way out from there.

00:05:27.660 --> 00:05:29.310
It might look
something like that.

00:05:29.310 --> 00:05:31.280
It might be way off the chart.

00:05:31.280 --> 00:05:33.922
Let's say the CEO made
$5 million in a year

00:05:33.922 --> 00:05:36.130
because he cashed in a bunch
of options or something.

00:05:36.130 --> 00:05:37.220
So it would be way over here.

00:05:37.220 --> 00:05:38.594
And maybe it's a
CEO and a couple

00:05:38.594 --> 00:05:41.740
of other people, the
CFO or the founders.

00:05:41.740 --> 00:05:45.480
So my guess is it
definitely wouldn't

00:05:45.480 --> 00:05:46.810
be a normal distribution.

00:05:46.810 --> 00:05:49.665
And it would be bimodal.

00:05:49.665 --> 00:05:52.165
You would have another peak
over here for senior management.

00:05:55.170 --> 00:05:57.560
Well, if we were
maybe in Europe,

00:05:57.560 --> 00:06:00.080
this would be
closer to the left.

00:06:00.080 --> 00:06:02.020
But it won't be a perfect
normal distribution.

00:06:02.020 --> 00:06:04.180
And you're not going
to have any values

00:06:04.180 --> 00:06:07.170
below a certain threshold,
below that kind of minimum wage

00:06:07.170 --> 00:06:07.670
level.

00:06:07.670 --> 00:06:09.130
So I would call
this, when you have

00:06:09.130 --> 00:06:11.340
a tail that goes
more to the right

00:06:11.340 --> 00:06:14.030
than to the left, a right
skewed distribution.

00:06:19.780 --> 00:06:22.340
Since it has two humps right
here, one there, and one there.

00:06:22.340 --> 00:06:23.585
We can also says it bimodal.

00:06:23.585 --> 00:06:25.710
I mean, it depends on what
kind of company this is.

00:06:25.710 --> 00:06:29.990
But that would be my guess of a
lot of large shipping company's

00:06:29.990 --> 00:06:31.060
salaries.

00:06:31.060 --> 00:06:35.150
Let's look at choice
C or problem part C.

00:06:35.150 --> 00:06:37.570
The annual salaries
of a random sample

00:06:37.570 --> 00:06:42.420
of 50 CEOs of major companies,
25 women and 25 men.

00:06:42.420 --> 00:06:43.930
The fact that they
wrote this year,

00:06:43.930 --> 00:06:46.540
I think they maybe are
implying that maybe men

00:06:46.540 --> 00:06:50.230
and women, the gender gap
has not been closed fully,

00:06:50.230 --> 00:06:52.730
and there is some discrepancy.

00:06:52.730 --> 00:06:57.050
So if I was just purely 50
CEOs of major companies,

00:06:57.050 --> 00:06:59.600
I would say it's probably
close to a normal distribution.

00:06:59.600 --> 00:07:02.940
It's probably something
like, once again, there's

00:07:02.940 --> 00:07:06.946
going to be some level below
which no CEO is willing to work

00:07:06.946 --> 00:07:08.695
for, although you have
heard of some cases

00:07:08.695 --> 00:07:09.580
where they work for free.

00:07:09.580 --> 00:07:11.400
But they're really getting
paid in other ways.

00:07:11.400 --> 00:07:12.858
If you include all
of those things,

00:07:12.858 --> 00:07:14.600
there's probably
some base salary

00:07:14.600 --> 00:07:17.090
that all CEOs make
at least that much.

00:07:17.090 --> 00:07:21.640
And then it goes up to some
value, the highest probability

00:07:21.640 --> 00:07:22.140
value.

00:07:22.140 --> 00:07:24.460
And then it probably has
a long tail to the right.

00:07:24.460 --> 00:07:26.370
And this is if there
were no gender gap.

00:07:26.370 --> 00:07:28.910
So this would just be a purely
right skewed distribution

00:07:28.910 --> 00:07:30.721
where you have a long
tail to the right.

00:07:30.721 --> 00:07:32.720
Now, if you assume that
there's some gender gap,

00:07:32.720 --> 00:07:34.830
then you might have
two humps here,

00:07:34.830 --> 00:07:36.480
which would be a
bimodal distribution.

00:07:36.480 --> 00:07:38.500
So if you assume
there's some gender gap,

00:07:38.500 --> 00:07:40.990
this is part C right
here, then maybe there's

00:07:40.990 --> 00:07:42.180
one hump for women.

00:07:42.180 --> 00:07:44.180
And if you assume that
women earn less than men,

00:07:44.180 --> 00:07:45.530
then another hump for men.

00:07:45.530 --> 00:07:47.540
And there are 25 of
each so there wouldn't

00:07:47.540 --> 00:07:49.450
necessarily more men than women.

00:07:49.450 --> 00:07:51.617
And then it would skew all
the way off to the right.

00:07:51.617 --> 00:07:53.741
And in fact, I think there
would probably be chance

00:07:53.741 --> 00:07:55.890
that you have this other
notion here where you have

00:07:55.890 --> 00:07:59.790
these super CEOs, or mega
CEOs who make millions,

00:07:59.790 --> 00:08:02.484
while most CEOs
probably just make,

00:08:02.484 --> 00:08:04.775
I'll put it in quotation
marks, "a few hundred thousand

00:08:04.775 --> 00:08:07.325
dollars" while there's
a small subset that

00:08:07.325 --> 00:08:10.060
are way off many standard
deviations to the right.

00:08:10.060 --> 00:08:13.580
So it could even be a
trimodal distribution here.

00:08:13.580 --> 00:08:16.450
So that's choice C. And
then so far, choice A

00:08:16.450 --> 00:08:18.490
looks like the best
candidate for a pure,

00:08:18.490 --> 00:08:21.210
or the closest to being
a normal distribution.

00:08:21.210 --> 00:08:23.290
Let's see what D is.

00:08:23.290 --> 00:08:26.640
The date of 100 pennies
taken from a cash drawer

00:08:26.640 --> 00:08:28.200
in a convenience store.

00:08:28.200 --> 00:08:29.130
100 pennies.

00:08:29.130 --> 00:08:31.470
So that's actually an
interesting experiment.

00:08:31.470 --> 00:08:33.500
But I would guess,
and once again, this

00:08:33.500 --> 00:08:35.470
is really a question
where I get to express

00:08:35.470 --> 00:08:38.239
my feelings about these things.

00:08:38.239 --> 00:08:39.780
As long as your
answer is reasonable,

00:08:39.780 --> 00:08:41.859
I would say that it is right.

00:08:41.859 --> 00:08:43.150
Most pennies are newer pennies.

00:08:43.150 --> 00:08:44.120
Because they go
out of commission.

00:08:44.120 --> 00:08:44.953
They get traded out.

00:08:44.953 --> 00:08:46.430
They get worn out as they age.

00:08:46.430 --> 00:08:50.844
They get lost, or they get
pressed at the little tourist

00:08:50.844 --> 00:08:52.510
place into those
little souvenir things.

00:08:52.510 --> 00:08:53.926
I'm not even sure
if that's legal,

00:08:53.926 --> 00:08:55.830
if you could do that
the money legally.

00:08:55.830 --> 00:08:59.820
So my guess is that if
you were to plot it,

00:08:59.820 --> 00:09:01.320
you would have a
ton of pennies that

00:09:01.320 --> 00:09:06.640
are, within the last few years,
So the date of 100 pennies, not

00:09:06.640 --> 00:09:10.550
their age, so the dates,
so if this is 2010,

00:09:10.550 --> 00:09:12.050
I would guess that
right now, you're

00:09:12.050 --> 00:09:14.020
not going to find
any 2010 pennies.

00:09:14.020 --> 00:09:18.780
But you're probably going to
find a ton of 2009 pennies.

00:09:18.780 --> 00:09:22.030
And then it probably just
goes down from there.

00:09:22.030 --> 00:09:24.010
And of course, you're
not going to find pennies

00:09:24.010 --> 00:09:26.765
that are older than
the United States

00:09:26.765 --> 00:09:28.640
or before they even
started printing pennies.

00:09:28.640 --> 00:09:30.420
So it's obvious this tail
isn't going to go to the left

00:09:30.420 --> 00:09:31.330
forever.

00:09:31.330 --> 00:09:33.200
But my guess is
you're going to have

00:09:33.200 --> 00:09:34.645
a left skewed distribution.

00:09:38.870 --> 00:09:41.710
Where you have the bulk of
the distribution on the right,

00:09:41.710 --> 00:09:43.550
but the tail goes
off to the left.

00:09:43.550 --> 00:09:45.633
That's why it's called a
left skewed distribution.

00:09:45.633 --> 00:09:51.950
Sometimes this is called a
negatively skewed distribution.

00:09:51.950 --> 00:09:55.040
And similarly, this right skewed
distribution, or this right you

00:09:55.040 --> 00:09:58.060
distribution, sometimes is
called positively skewed.

00:09:58.060 --> 00:09:59.950
And if you have only one hump.

00:09:59.950 --> 00:10:03.100
You don't have a multimodal
distribution like this,

00:10:03.100 --> 00:10:05.610
in a left skewed
distribution, your mean

00:10:05.610 --> 00:10:08.334
is going to be to the
left of your median.

00:10:08.334 --> 00:10:09.750
So in this case,
maybe your median

00:10:09.750 --> 00:10:11.090
might be someplace over here.

00:10:11.090 --> 00:10:13.770
But since you have this
long tail to the left,

00:10:13.770 --> 00:10:16.790
your mean might be
someplace over here.

00:10:16.790 --> 00:10:18.410
And likewise, in
this distribution,

00:10:18.410 --> 00:10:22.006
your median, your middle value,
might be some place like this.

00:10:22.006 --> 00:10:24.380
But because it's right skewed,
and for the most part only

00:10:24.380 --> 00:10:25.710
has one big hump.

00:10:25.710 --> 00:10:28.147
this hump won't change things
too much because it's small,

00:10:28.147 --> 00:10:29.980
your mean is going to
be to the right of it.

00:10:29.980 --> 00:10:31.355
So that's another
reason why it's

00:10:31.355 --> 00:10:35.290
called a right skewed or
positively skewed distribution.

00:10:35.290 --> 00:10:37.300
So to answer the
question, you know,

00:10:37.300 --> 00:10:38.970
these are my feelings
about all of them.

00:10:38.970 --> 00:10:40.762
But I would say--
the other choices

00:10:40.762 --> 00:10:42.720
explain why you believe
they would not follow--

00:10:42.720 --> 00:10:44.636
or they said, which of
the following data sets

00:10:44.636 --> 00:10:47.070
is most likely to be
normally distributed.

00:10:47.070 --> 00:10:49.890
Well, I would say choice
A. But it's really

00:10:49.890 --> 00:10:52.880
a matter of opinion, at
least in this question.


MTEL Math Practice Test: 31-35

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:02.570
Problem 31.

00:00:02.570 --> 00:00:04.180
Use the problem below
to answer the

00:00:04.180 --> 00:00:05.440
question that follows.

00:00:05.440 --> 00:00:06.820
All right, so this
is our problem.

00:00:06.820 --> 00:00:10.420
A red car and a blue car compete
in two 100 mile races.

00:00:10.420 --> 00:00:12.490
That's the first race, that's
the second race.

00:00:12.490 --> 00:00:15.130
In the first race, both cars
leave the starting

00:00:15.130 --> 00:00:16.300
line at the same time.

00:00:16.300 --> 00:00:17.480
That's right there.

00:00:17.480 --> 00:00:20.770
When the red car crosses the
finish line, the blue car has

00:00:20.770 --> 00:00:22.500
10 miles left to go.

00:00:22.500 --> 00:00:23.510
They drew that right there.

00:00:23.510 --> 00:00:27.080
Red car crosses the finish line,
blue car has 10 miles

00:00:27.080 --> 00:00:28.140
left to go.

00:00:28.140 --> 00:00:31.425
In the second race, both cars
start at the same time.

00:00:31.425 --> 00:00:34.150
But while the blue car begins
at the starting line, right

00:00:34.150 --> 00:00:37.000
there, the red car begins
10 miles behind

00:00:37.000 --> 00:00:37.980
the starting line.

00:00:37.980 --> 00:00:38.900
Fair enough.

00:00:38.900 --> 00:00:41.640
Assuming that each car's
average speed does not

00:00:41.640 --> 00:00:43.250
change-- that's a
big assumption.

00:00:43.250 --> 00:00:45.570
So I'm assuming from race to
race, they have the same

00:00:45.570 --> 00:00:46.960
average speed.

00:00:46.960 --> 00:00:49.860
How far has the blue car
traveled in the second race

00:00:49.860 --> 00:00:51.660
when the red car reaches
the finish line?

00:00:51.660 --> 00:00:54.380
So the red car reaches the
finish line, the blue car has

00:00:54.380 --> 00:00:56.300
traveled this distance x.

00:00:56.300 --> 00:00:58.550
That's what they want us
to solve for, for x.

00:00:58.550 --> 00:01:00.900
How far has the blue car
traveled with the red car

00:01:00.900 --> 00:01:03.730
finishes the finish line?

00:01:03.730 --> 00:01:06.580
Now, this statement right here,
assuming that each car's

00:01:06.580 --> 00:01:09.170
average speed, at least in
my brain, is a big clue.

00:01:09.170 --> 00:01:11.760
It tells us that that's
important-- that their average

00:01:11.760 --> 00:01:12.640
speed is the same.

00:01:12.640 --> 00:01:14.930
So from race to race, they don't
have, obviously, the

00:01:14.930 --> 00:01:15.840
same average speed.

00:01:15.840 --> 00:01:18.310
The red car is faster
in either case.

00:01:18.310 --> 00:01:24.130
So let's say that r is equal
to the red's speed.

00:01:24.130 --> 00:01:30.370
And let's say that blue is
equal to blue's speed.

00:01:30.370 --> 00:01:34.030
Now, in either situation, either
drawing, let's say in

00:01:34.030 --> 00:01:36.350
the first race, they
both traveled the

00:01:36.350 --> 00:01:37.850
same amount of time.

00:01:37.850 --> 00:01:39.760
The same amount of time it took
the red car to go the

00:01:39.760 --> 00:01:43.170
whole 100 miles, the blue car
traveled this distance, which

00:01:43.170 --> 00:01:44.110
is 90 miles.

00:01:44.110 --> 00:01:45.115
How did I know 90 miles?

00:01:45.115 --> 00:01:48.720
Because he has 10 miles left
to go in a 100 mile race.

00:01:48.720 --> 00:01:52.130
So if we know that distance is
equal to-- sometimes you say

00:01:52.130 --> 00:01:54.980
speed times time, or rate times
time; I'll use speed,

00:01:54.980 --> 00:01:56.870
just because I already
used r-- is equal

00:01:56.870 --> 00:01:58.400
to speed times time.

00:01:58.400 --> 00:02:00.480
That also tells us that
time is equal to

00:02:00.480 --> 00:02:03.130
distance divided by speed.

00:02:03.130 --> 00:02:10.320
So in the first race, in race
1, the time that the red car

00:02:10.320 --> 00:02:19.310
took to finish the 100 miles
is 100 miles divided by r

00:02:19.310 --> 00:02:22.740
miles per hour.

00:02:22.740 --> 00:02:25.385
And notice, the units will
cancel out properly.

00:02:25.385 --> 00:02:28.810
If you divide miles by miles
per hour, you're left just

00:02:28.810 --> 00:02:29.560
with hours.

00:02:29.560 --> 00:02:33.510
So this is the time it takes
for the first car

00:02:33.510 --> 00:02:35.200
to finish the race.

00:02:35.200 --> 00:02:37.570
Now, we also know that's the
same amount of time it took

00:02:37.570 --> 00:02:39.330
the blue car to go 90 miles.

00:02:39.330 --> 00:02:48.300
So that is going to be equal to
90 miles over b, or blue's

00:02:48.300 --> 00:02:52.380
miles per hour.

00:02:52.380 --> 00:02:52.650
Right?

00:02:52.650 --> 00:02:54.130
That's what the first
race tells us.

00:02:54.130 --> 00:02:55.990
Now, what does the second
race tell us?

00:02:55.990 --> 00:02:59.320
The second race, we can
do the same logic.

00:02:59.320 --> 00:03:01.790
Race 2.

00:03:01.790 --> 00:03:06.180
So, in the time it takes the red
car to travel 110 miles.

00:03:06.180 --> 00:03:06.350
Right?

00:03:06.350 --> 00:03:09.640
He drives the whole 100 miles,
and he started 10 miles back.

00:03:09.640 --> 00:03:18.580
So the red car, 110 miles
divided by red miles per hour

00:03:18.580 --> 00:03:21.660
is equal to what?

00:03:21.660 --> 00:03:23.820
It's equal to x.

00:03:23.820 --> 00:03:29.070
The distance the blue car
travels is equal to x miles

00:03:29.070 --> 00:03:36.190
divided by blue's miles
per hour, b miles per

00:03:36.190 --> 00:03:38.360
hour, just like that.

00:03:38.360 --> 00:03:39.780
Now, what do we have here?

00:03:39.780 --> 00:03:44.460
We want to solve for x, and
maybe, just maybe, if we can

00:03:44.460 --> 00:03:48.360
find an expression for the ratio
between r and b, then

00:03:48.360 --> 00:03:49.540
maybe we can solve for x.

00:03:49.540 --> 00:03:50.880
Let me show you what
I mean by that.

00:03:50.880 --> 00:03:52.900
Let me switch colors to make
this a little bit less

00:03:52.900 --> 00:03:53.820
monotonous.

00:03:53.820 --> 00:03:57.570
So this first, from race 1, we
got the information that 100

00:03:57.570 --> 00:04:04.380
over r is equal to 90 over b.

00:04:04.380 --> 00:04:08.400
If we cross multiply, we
essentially get 100b, 100

00:04:08.400 --> 00:04:12.280
times b is equal
to 90 times r.

00:04:12.280 --> 00:04:16.890
If we divide both sides by
90, we get 100 over 90

00:04:16.890 --> 00:04:18.295
b is equal to r.

00:04:18.295 --> 00:04:22.580
If you divide both sides by b,
and you swap the sides, you

00:04:22.580 --> 00:04:27.130
get r over b is equal
to 100 over 90.

00:04:27.130 --> 00:04:30.480
So we know what the ratio of
red speed to blue speed is.

00:04:30.480 --> 00:04:32.620
It's 100 to 90.

00:04:32.620 --> 00:04:34.820
Now, can we do the
same thing here?

00:04:34.820 --> 00:04:36.070
Let me do this in red.

00:04:38.410 --> 00:04:41.800
OK, so here, if we cross
multiply, from race 2, we have

00:04:41.800 --> 00:04:45.800
110 over r is equal
to x over b.

00:04:45.800 --> 00:04:52.170
If we cross multiply, we get
110b is equal to rx.

00:04:52.170 --> 00:04:55.160
And then, let's see, if we
divide both sides by x, we get

00:04:55.160 --> 00:04:59.670
110 over x b is equal to r.

00:04:59.670 --> 00:05:04.340
And then if we divide both sides
by b, we get 110 over x

00:05:04.340 --> 00:05:07.250
is equal to r over b.

00:05:07.250 --> 00:05:09.660
That's the information that
the second race gave us.

00:05:09.660 --> 00:05:12.130
Now, the first race already
told us what r

00:05:12.130 --> 00:05:13.180
over b is equal to.

00:05:13.180 --> 00:05:16.550
It's equal to 100 over 90.

00:05:16.550 --> 00:05:19.460
So we can replace r over b,
based on the information in

00:05:19.460 --> 00:05:26.710
the first race, with
100 over 90.

00:05:26.710 --> 00:05:29.720
Now, let's see, so our big
takeaway-- notice, they don't

00:05:29.720 --> 00:05:31.690
want us to solve for x, they
just want us to set up the

00:05:31.690 --> 00:05:39.370
problem-- is 110 over x is
equal to 100 over 90.

00:05:39.370 --> 00:05:41.730
Now, let's see which of these
choices say that.

00:05:41.730 --> 00:05:46.410
110 over x is equal
to 100 over 90.

00:05:46.410 --> 00:05:47.600
That's choice A.

00:05:47.600 --> 00:05:49.510
Just like that.

00:05:49.510 --> 00:05:50.760
Next problem.

00:05:53.340 --> 00:05:56.100
All right.

00:05:56.100 --> 00:05:58.080
Use the diagram below
to answer the

00:05:58.080 --> 00:05:59.810
question that follows.

00:05:59.810 --> 00:06:02.570
The diagram above is used to
describe the relationship

00:06:02.570 --> 00:06:05.220
between the circumference--
right, the circumference is

00:06:05.220 --> 00:06:09.310
going around the circle-- the
radius-- that's the radius--

00:06:09.310 --> 00:06:10.595
and the area of a circle.

00:06:10.595 --> 00:06:12.400
The area, we know what
the area is.

00:06:12.400 --> 00:06:14.770
Assuming that the circle is
divided into enough sections

00:06:14.770 --> 00:06:16.770
so that the figure on the
right approximates a

00:06:16.770 --> 00:06:20.030
rectangle, which of the
following relationships is

00:06:20.030 --> 00:06:21.220
demonstrated?

00:06:21.220 --> 00:06:24.380
So here they divided
into six wedges.

00:06:24.380 --> 00:06:26.320
It looks like a Trivial Pursuit
piece, and they put

00:06:26.320 --> 00:06:26.950
them all up.

00:06:26.950 --> 00:06:29.020
But you could imagine, if you
made the wedges even thinner,

00:06:29.020 --> 00:06:31.870
if you divided into 12 wedges,
then it would start

00:06:31.870 --> 00:06:32.990
to look like this.

00:06:32.990 --> 00:06:36.690
It would look like this if you
did it into 12 wedges.

00:06:36.690 --> 00:06:38.870
And if you did it into 24
wedges, it would start looking

00:06:38.870 --> 00:06:42.690
more and more like a rectangle
that looks

00:06:42.690 --> 00:06:44.310
something like that.

00:06:44.310 --> 00:06:46.340
The more you divide the
wedges, it looks more

00:06:46.340 --> 00:06:47.190
and more like that.

00:06:47.190 --> 00:06:48.860
So let's see what they're
trying to show us.

00:06:48.860 --> 00:06:51.110
So if this is r, right here.

00:06:51.110 --> 00:06:53.960
On the wedges, this distance--
or actually this

00:06:53.960 --> 00:06:56.620
distance-- is also r.

00:06:56.620 --> 00:06:56.960
Right?

00:06:56.960 --> 00:06:59.530
That distance is also r.

00:06:59.530 --> 00:07:01.950
The whole thing is obviously
the circumference.

00:07:01.950 --> 00:07:05.850
Over here, on our kind of
approximated rectangle, what's

00:07:05.850 --> 00:07:07.070
our circumference?

00:07:07.070 --> 00:07:09.640
Well, our circumference is that
plus that, plus that,

00:07:09.640 --> 00:07:12.430
plus that, plus that,
plus that.

00:07:12.430 --> 00:07:15.300
So if we just look at maybe
the base of our rectangle,

00:07:15.300 --> 00:07:20.180
that is 1/2 our circumference.

00:07:20.180 --> 00:07:20.550
Right?

00:07:20.550 --> 00:07:22.450
That's 1/2 our circumference.

00:07:22.450 --> 00:07:26.260
Because our circumference is
these arcs plus this arc,

00:07:26.260 --> 00:07:27.380
these three arcs.

00:07:27.380 --> 00:07:28.860
This is just 1/2 of them.

00:07:28.860 --> 00:07:32.840
So the area of the rectangle,
the area of this rectangle--

00:07:32.840 --> 00:07:36.350
which is the same thing as the
area of the circle, because

00:07:36.350 --> 00:07:38.960
all we did is we just kind
of cut up the circle and

00:07:38.960 --> 00:07:41.300
rearranged it into a rectangle,
a reapproximated

00:07:41.300 --> 00:07:44.020
rectangle-- the area of the
circle of the rectangle is

00:07:44.020 --> 00:07:48.010
going to be the base, which
is equal to 1/2 times the

00:07:48.010 --> 00:07:51.770
circumference, times the height,
which is the radius.

00:07:54.550 --> 00:07:56.870
And, well, lucky for me, that's
actually the actual

00:07:56.870 --> 00:07:59.450
choice, choice A.

00:07:59.450 --> 00:08:00.410
Just like that.

00:08:00.410 --> 00:08:02.350
Actually, this is
a fun exercise.

00:08:02.350 --> 00:08:07.020
Maybe I'll do a whole video
explaining why area is 1/2 the

00:08:07.020 --> 00:08:09.240
radius times the circumference
using this example.

00:08:09.240 --> 00:08:13.650
Because if you divide it into
more and more and more wedges,

00:08:13.650 --> 00:08:16.010
it's going to look more and
more like a rectangle.

00:08:16.010 --> 00:08:18.440
I'll leave that for you
to experiment with.

00:08:18.440 --> 00:08:21.120
Problem 33.

00:08:21.120 --> 00:08:24.090
A pretzel company sells pretzels
in a cylindrical

00:08:24.090 --> 00:08:26.820
container with a radius
of 10 centimeters

00:08:26.820 --> 00:08:27.860
and a height of 30.

00:08:27.860 --> 00:08:29.580
Let me draw it.

00:08:29.580 --> 00:08:32.750
So a cylindrical container,
radius is 10 centimeters, and

00:08:32.750 --> 00:08:38.240
then the height of our container
is 30 centimeters.

00:08:38.240 --> 00:08:38.520
Right?

00:08:38.520 --> 00:08:39.190
It's all centimeters.

00:08:39.190 --> 00:08:42.380
The company's packaging
designers are considering

00:08:42.380 --> 00:08:45.590
switching to a new cylindrical
container with a radius of

00:08:45.590 --> 00:08:49.600
20-- so it's a bigger radius,
radius of 20 centimeters-- and

00:08:49.600 --> 00:08:54.680
a height of 15 centimeters.

00:08:54.680 --> 00:08:57.750
How does the volume of the
proposed new container compare

00:08:57.750 --> 00:09:00.050
to the volume of the
old container?

00:09:00.050 --> 00:09:03.270
So the old container's volume is
going to be this area times

00:09:03.270 --> 00:09:03.910
the height.

00:09:03.910 --> 00:09:05.940
So it's going to be, so
the volume is equal to

00:09:05.940 --> 00:09:08.570
the area pi r squared.

00:09:08.570 --> 00:09:09.800
r is 10.

00:09:09.800 --> 00:09:11.650
r squared is 100.

00:09:11.650 --> 00:09:15.810
So it's going to be 100pi,
that's the area up here.

00:09:15.810 --> 00:09:18.250
That's 100pi.

00:09:18.250 --> 00:09:19.780
And then we're going to multiply
that times the

00:09:19.780 --> 00:09:23.770
height, times 30 centimeters,
and so it's going to be equal

00:09:23.770 --> 00:09:29.180
to 3,000pi cubic centimeters.

00:09:29.180 --> 00:09:30.620
That's our old container.

00:09:30.620 --> 00:09:32.800
Our new container-- what's
the area up here?

00:09:32.800 --> 00:09:34.700
Pi r squared.

00:09:34.700 --> 00:09:38.340
The area is equal to
pi times r squared.

00:09:38.340 --> 00:09:39.450
r is 20.

00:09:39.450 --> 00:09:42.330
20 squared is 400.

00:09:42.330 --> 00:09:49.980
So our volume is equal
to 400pi times 15.

00:09:49.980 --> 00:09:52.820
And what's 400 times 15?

00:09:52.820 --> 00:09:54.280
400 times 15 is what?

00:09:54.280 --> 00:09:55.440
6,000.

00:09:55.440 --> 00:09:57.930
400 times 10 is 4,000.

00:09:57.930 --> 00:10:00.120
400 times 5 is 2,000.

00:10:00.120 --> 00:10:00.270
Right.

00:10:00.270 --> 00:10:01.190
It's 6,000.

00:10:01.190 --> 00:10:06.870
So the volume of our new
container is 6,000pi

00:10:06.870 --> 00:10:08.545
centimeters, or cubic
centimeters.

00:10:08.545 --> 00:10:10.930
So our new container
has twice the

00:10:10.930 --> 00:10:12.740
volume of our old container.

00:10:12.740 --> 00:10:14.380
So let's see if they say that.

00:10:14.380 --> 00:10:14.690
Let's see.

00:10:14.690 --> 00:10:20.450
The volume of the new container
is twice the volume

00:10:20.450 --> 00:10:21.670
of the old container.

00:10:21.670 --> 00:10:23.470
That is correct.

00:10:23.470 --> 00:10:24.970
Choice D.

00:10:24.970 --> 00:10:26.220
Next problem.

00:10:29.320 --> 00:10:31.700
All right.

00:10:31.700 --> 00:10:35.790
A fuel tank was approximately
1/8 full.

00:10:35.790 --> 00:10:41.890
After adding $50 worth of fuel,
the tank was 3/4 full.

00:10:41.890 --> 00:10:45.890
If the fuel cost p dollars
per gallon, all right,

00:10:45.890 --> 00:10:50.600
approximately how many gallons
does the tank hold when full?

00:10:50.600 --> 00:10:54.100
So, let's see, we added $50, and
it's p dollars per gallon,

00:10:54.100 --> 00:11:02.690
so we added-- so 50 divided by
p gallons took us from 1/8 to

00:11:02.690 --> 00:11:05.380
3/4 full of a tank.

00:11:05.380 --> 00:11:07.300
So how much did it add?

00:11:07.300 --> 00:11:12.200
We went from 3/4 of a tank--
from 1/8 of a tank to 3/4.

00:11:12.200 --> 00:11:13.380
So how much did we add?

00:11:13.380 --> 00:11:15.760
If we just take the difference
between the two, we can

00:11:15.760 --> 00:11:21.840
rewrite 3/4 as 6/8, minus
1/8, equal to 5/8.

00:11:21.840 --> 00:11:23.790
So we added 5/8 of a tank.

00:11:23.790 --> 00:11:27.520
Going from 1/8 to 3/4 is
adding 5/8 of a tank.

00:11:27.520 --> 00:11:34.180
So this is equal to 5/8 of
a tank, of a tank of

00:11:34.180 --> 00:11:35.870
gas, of our fuel tank.

00:11:35.870 --> 00:11:37.930
Now they're saying,
approximately how many gallons

00:11:37.930 --> 00:11:39.430
does the tank hold when full?

00:11:39.430 --> 00:11:41.420
And it's all in terms of p.

00:11:41.420 --> 00:11:44.620
So to solve for one tank, we
can multiply both sides of

00:11:44.620 --> 00:11:48.940
this equation times the
reciprocal of this, times 8/5.

00:11:48.940 --> 00:11:50.400
Let's do that.

00:11:50.400 --> 00:11:54.380
So you multiply 8 over 5 on the
left-hand side, times 8

00:11:54.380 --> 00:11:56.290
over 5 on the right-hand side.

00:11:56.290 --> 00:11:59.950
And then you're left with--
well, that can just cancel.

00:11:59.950 --> 00:12:01.380
5's cancel here.

00:12:01.380 --> 00:12:02.230
8's cancel here.

00:12:02.230 --> 00:12:02.960
That's why I did it.

00:12:02.960 --> 00:12:09.180
So one tank is equal to 8 over
5, times 50 over p gallons.

00:12:09.180 --> 00:12:11.490
So if we just divide the
numerator and the denominator

00:12:11.490 --> 00:12:13.270
by 5, it becomes 10.

00:12:13.270 --> 00:12:14.710
That becomes 1.

00:12:14.710 --> 00:12:22.930
8 times 10 is 80 over p gallons,
is equal to 1 tank.

00:12:22.930 --> 00:12:25.575
And that is choice A.

00:12:28.300 --> 00:12:29.860
Next question.

00:12:29.860 --> 00:12:32.870
A homeowner-- let me switch
colors-- a homeowner is

00:12:32.870 --> 00:12:36.160
planning to use carpet tiles to
cover the floor of a room

00:12:36.160 --> 00:12:40.160
measuring 9 feet by
10 feet, 8 inches.

00:12:40.160 --> 00:12:48.170
So it's 9 feet by 10
feet, 8 inches.

00:12:48.170 --> 00:12:51.790
Whenever I see that, I like to
write it as a decimal, or I

00:12:51.790 --> 00:12:52.870
like to write it just
purely as inches.

00:12:52.870 --> 00:12:55.030
How many inches is this?

00:12:55.030 --> 00:12:57.200
10 feet is 120 inches, right?

00:12:57.200 --> 00:12:58.260
12 inches per foot.

00:12:58.260 --> 00:12:59.910
So 120 plus 8.

00:12:59.910 --> 00:13:04.380
So this is also 128 inches.

00:13:04.380 --> 00:13:05.210
So I'll just write that there.

00:13:05.210 --> 00:13:08.970
So it's 9 feet by 128 inches.

00:13:08.970 --> 00:13:13.120
If the carpet tiles are 8 inches
wide and 1 foot long,

00:13:13.120 --> 00:13:15.430
and there are no gaps between
the tiles when they're placed

00:13:15.430 --> 00:13:18.090
on the floor, how many carpet
tiles will the homeowner need

00:13:18.090 --> 00:13:20.340
to cover the floor
of the room?

00:13:20.340 --> 00:13:22.100
So I'm going to place them this
way, just because I kept

00:13:22.100 --> 00:13:23.200
this side in foot.

00:13:23.200 --> 00:13:26.200
So I'm going to use the foot
dimension on the foot side.

00:13:26.200 --> 00:13:29.630
So this is 1 foot by 8 inches.

00:13:29.630 --> 00:13:30.300
Right?

00:13:30.300 --> 00:13:31.710
That's the dimension
of our tile.

00:13:31.710 --> 00:13:33.400
Let me do it in a
different color.

00:13:33.400 --> 00:13:36.440
It's 1 foot on this side
right here, and

00:13:36.440 --> 00:13:37.800
then 8 inches on that.

00:13:37.800 --> 00:13:39.470
So clearly, we're going to be
able to add-- we're going to

00:13:39.470 --> 00:13:42.060
have 9 tiles, because
we have 9 feet.

00:13:42.060 --> 00:13:44.840
So we're going to have 9 tiles
in this direction.

00:13:44.840 --> 00:13:47.190
And then how many tiles
in this direction?

00:13:47.190 --> 00:13:50.400
It's 8 inches, and we have
a total of 128 inches.

00:13:50.400 --> 00:13:53.830
So how many times does
8 to go into 128?

00:13:53.830 --> 00:13:57.740
8 goes into 12 once,
1 times 8 is 8, 48.

00:13:57.740 --> 00:14:00.160
8 goes into 48 six times.

00:14:00.160 --> 00:14:03.260
So you'll have 16 tiles
in that direction.

00:14:03.260 --> 00:14:08.310
9 tiles here in the, I guess,
on the width, 16

00:14:08.310 --> 00:14:09.550
tiles in the length.

00:14:09.550 --> 00:14:12.960
So we're going to have
16 times 9 tiles.

00:14:12.960 --> 00:14:16.420
6 times 9 is 54.

00:14:16.420 --> 00:14:21.050
1 times 9 is 9, plus 5 is 14.

00:14:21.050 --> 00:14:23.465
So we're going to
need 144 tiles.

00:14:23.465 --> 00:14:26.060
So that's choice C.


MTEL Math Practice Test: 36-40

WEBVTT
Kind: captions
Language: en

00:00:00.440 --> 00:00:02.430
Problem 36.

00:00:02.430 --> 00:00:04.800
Use the diagram below
to answer the

00:00:04.800 --> 00:00:05.690
question that follows.

00:00:05.690 --> 00:00:06.210
All right.

00:00:06.210 --> 00:00:07.530
There's my diagram.

00:00:07.530 --> 00:00:12.790
A gift box has dimensions
x by y by z.

00:00:12.790 --> 00:00:16.410
A decorative ribbon is wrapped
across the diagonals of the

00:00:16.410 --> 00:00:18.210
box, as shown above.

00:00:18.210 --> 00:00:20.550
Which of the following
expressions represents the

00:00:20.550 --> 00:00:24.190
approximate total length
of the ribbon?

00:00:24.190 --> 00:00:24.840
All right.

00:00:24.840 --> 00:00:27.000
So let's just measure
this ribbon.

00:00:27.000 --> 00:00:29.630
So what's this length,
right here?

00:00:29.630 --> 00:00:31.390
What's that length,
right there?

00:00:31.390 --> 00:00:36.290
Well if we draw that side of the
box, the height is z, and

00:00:36.290 --> 00:00:38.320
the depth is y.

00:00:38.320 --> 00:00:42.775
And the ribbon goes from the
bottom left to the top right.

00:00:45.770 --> 00:00:48.720
So we can use the Pythagorean
theorem right here, right?

00:00:48.720 --> 00:00:51.610
Because if this height is
z, then this height,

00:00:51.610 --> 00:00:52.910
right here, is z.

00:00:52.910 --> 00:00:56.170
So this is a right triangle, and
so we can figure out the

00:00:56.170 --> 00:00:58.230
hypotenuse using the Pythagorean
theorem.

00:00:58.230 --> 00:01:01.500
So we could say that the length,
right here, the length

00:01:01.500 --> 00:01:05.360
squared-- maybe I'll write it
like that-- the length squared

00:01:05.360 --> 00:01:09.240
is equal to y squared
plus z squared.

00:01:09.240 --> 00:01:12.770
Or the length is equal to
the square root of y

00:01:12.770 --> 00:01:14.980
squared plus z squared.

00:01:14.980 --> 00:01:16.830
That's this length right here.

00:01:16.830 --> 00:01:20.030
Now, we could use that exact
same argument on this side of

00:01:20.030 --> 00:01:22.030
the box, over here.

00:01:22.030 --> 00:01:24.600
You can imagine just, if we were
to face it-- well, if you

00:01:24.600 --> 00:01:26.150
were to just turn it around,
it would look

00:01:26.150 --> 00:01:27.860
exactly like this.

00:01:27.860 --> 00:01:31.870
Or if you were to look at it
from the inside of the box, it

00:01:31.870 --> 00:01:33.570
would look like this.

00:01:33.570 --> 00:01:35.600
It looks like this is
the back of it.

00:01:35.600 --> 00:01:37.760
So if we're looking at it
from this direction,

00:01:37.760 --> 00:01:38.720
it looks like this.

00:01:38.720 --> 00:01:40.650
The height is z.

00:01:40.650 --> 00:01:42.680
The base is y.

00:01:42.680 --> 00:01:44.330
And so we just have
the hypotenuse.

00:01:44.330 --> 00:01:47.240
y squared plus z squared is
going to be equal to the

00:01:47.240 --> 00:01:48.360
length squared.

00:01:48.360 --> 00:01:51.330
Or the length is equal to y
squared plus the square root

00:01:51.330 --> 00:01:53.300
of y squared, plus z squared.

00:01:53.300 --> 00:01:55.670
So that's this length
and that length.

00:01:55.670 --> 00:01:58.160
So, so far, if we just wanted to
add up just that length and

00:01:58.160 --> 00:02:02.310
that length, we're going to
have 2 times each of them.

00:02:02.310 --> 00:02:06.590
So 2 times the square of y
squared, plus z squared.

00:02:06.590 --> 00:02:08.930
That's the two red parts
of the ribbon.

00:02:08.930 --> 00:02:13.690
Now, what is this length
right here?

00:02:13.690 --> 00:02:14.730
Well, same argument.

00:02:14.730 --> 00:02:18.550
If I were to just draw the base
of this rectangle, the

00:02:18.550 --> 00:02:22.230
depth is y, the width is x,
and the ribbon just goes

00:02:22.230 --> 00:02:23.520
across it like that.

00:02:26.780 --> 00:02:31.060
So if we call this the length,
l, of this blue part, you have

00:02:31.060 --> 00:02:35.280
l squared is equal to x squared
plus y squared, or l

00:02:35.280 --> 00:02:39.610
is equal to the square root of
x squared plus y squared.

00:02:39.610 --> 00:02:41.610
That's this length,
right there.

00:02:41.610 --> 00:02:44.370
And then this length over here
might not look the same,

00:02:44.370 --> 00:02:46.965
because of the way that the
perspective works, but it's

00:02:46.965 --> 00:02:48.580
the exact same thing.

00:02:48.580 --> 00:02:49.820
It's the exact same thing.

00:02:49.820 --> 00:02:52.730
If this is x, this depth is y.

00:02:52.730 --> 00:02:55.720
If you were look at the box from
the top, this length is

00:02:55.720 --> 00:02:58.740
x, the depth is y, the top
ribbon looks like that.

00:02:58.740 --> 00:03:00.880
So x squared plus y squared
is going to be

00:03:00.880 --> 00:03:01.660
equal to that thing.

00:03:01.660 --> 00:03:04.360
So this length is also
the square root of x

00:03:04.360 --> 00:03:06.590
squared plus y squared.

00:03:06.590 --> 00:03:08.770
So that's the square root of
x squared plus y squared.

00:03:08.770 --> 00:03:10.750
This is the square root of
x squared plus y squared.

00:03:10.750 --> 00:03:12.890
So in the total length of the
ribbon, we're going to have

00:03:12.890 --> 00:03:13.750
two of them.

00:03:13.750 --> 00:03:18.380
Plus 2 times the square root of
x squared plus y squared.

00:03:18.380 --> 00:03:21.400
So the total length of the
ribbon: 2 times the square

00:03:21.400 --> 00:03:24.540
root of x squared, plus y
squared, plus 2 times the

00:03:24.540 --> 00:03:28.670
square root of y squared,
plus z squared.

00:03:28.670 --> 00:03:32.060
Let's see, that looks
like choice C.

00:03:32.060 --> 00:03:36.130
They just factored the 2 out,
so it's 2 times everything.

00:03:36.130 --> 00:03:39.840
Square root of x squared plus
y squared, plus the square

00:03:39.840 --> 00:03:43.170
root of y squared
plus z squared.

00:03:43.170 --> 00:03:44.810
Right over there.

00:03:44.810 --> 00:03:46.860
Next problem.

00:03:46.860 --> 00:03:48.280
Problem 37.

00:03:48.280 --> 00:03:51.890
Use the graph below to answer
the question that follows.

00:03:51.890 --> 00:03:52.300
All right.

00:03:52.300 --> 00:03:54.640
We have this little drawing
of a hand pointing up.

00:03:54.640 --> 00:03:57.140
If the image of a pointing hand
in the graph above is

00:03:57.140 --> 00:04:01.460
rotated 180 degrees about the
origin, and then reflected

00:04:01.460 --> 00:04:03.400
across the x-axis, which of the

00:04:03.400 --> 00:04:05.420
following graphs will result?

00:04:05.420 --> 00:04:06.590
So let's do it step by step.

00:04:06.590 --> 00:04:11.000
If you rotate it 180 degrees,
you're rotating it like this.

00:04:11.000 --> 00:04:13.610
Around the x-axis, you're
rotating it like that.

00:04:13.610 --> 00:04:17.610
So for example, this square--
we could draw it, maybe I'll

00:04:17.610 --> 00:04:20.579
do it in a lighter color-- this
right here, this part of

00:04:20.579 --> 00:04:22.690
the hand is going to rotate it
all the way down, so it's

00:04:22.690 --> 00:04:26.480
going to look like that,
right there, if I

00:04:26.480 --> 00:04:27.820
were to shade it in.

00:04:27.820 --> 00:04:30.690
And then the hand-- this part,
right here, is going to be

00:04:30.690 --> 00:04:33.660
rotated all the way
down like this.

00:04:33.660 --> 00:04:35.780
So the hand's going to look
something like this.

00:04:35.780 --> 00:04:37.970
That's the thumb part.

00:04:37.970 --> 00:04:40.350
And then you're going to have
the finger coming out like

00:04:40.350 --> 00:04:43.530
this, the finger part, the
finger pointing up.

00:04:43.530 --> 00:04:44.850
And then you're going to
have the other fingers.

00:04:44.850 --> 00:04:46.650
So the hand's going to look
something like that when you

00:04:46.650 --> 00:04:47.810
rotate it 180 degrees.

00:04:47.810 --> 00:04:50.410
I just rotated around
the origin.

00:04:50.410 --> 00:04:52.770
Now, the second part
is that then,

00:04:52.770 --> 00:04:55.390
reflected across the x-axis.

00:04:55.390 --> 00:04:58.900
So then if we reflect this
across the x-axis, it's going

00:04:58.900 --> 00:04:59.660
to look like this.

00:04:59.660 --> 00:05:03.500
It's going to look something
like that, like right there.

00:05:03.500 --> 00:05:04.700
So that should be our answer.

00:05:04.700 --> 00:05:07.570
So which of the choices
looks like that?

00:05:07.570 --> 00:05:12.520
Looks like choice C.

00:05:12.520 --> 00:05:14.910
Choice C looks exactly
what we drew up here.

00:05:14.910 --> 00:05:19.760
Well, the finger is closer to
the vertical axis than, I

00:05:19.760 --> 00:05:22.650
guess, the pinky is in either
case, because we rotated

00:05:22.650 --> 00:05:24.550
around and then we flipped.

00:05:24.550 --> 00:05:27.110
And that is choice C.

00:05:27.110 --> 00:05:28.360
Next question.

00:05:32.890 --> 00:05:34.510
Problem 38.

00:05:34.510 --> 00:05:36.930
Which of the following
nets can be folded to

00:05:36.930 --> 00:05:38.780
form a square pyramid?

00:05:38.780 --> 00:05:40.820
So a square pyramid-- this is
all, you're going to get some

00:05:40.820 --> 00:05:43.350
visualization problem-- a
square pyramid is just a

00:05:43.350 --> 00:05:45.830
pyramid that has a square
as its base, like

00:05:45.830 --> 00:05:47.540
the pyramids in Egypt.

00:05:47.540 --> 00:05:50.840
So it has a square
as its base.

00:05:50.840 --> 00:05:51.930
So that's the square base.

00:05:51.930 --> 00:05:53.880
Let's see if I can
construct one.

00:05:53.880 --> 00:05:59.410
If I were to fold this guy up,
and then this guy up, then can

00:05:59.410 --> 00:06:01.570
this guy be the other
side of the pyramid?

00:06:01.570 --> 00:06:03.660
No, because if I fold this
guy up like this.

00:06:03.660 --> 00:06:05.460
So let's say, imagine,
let's fold them up.

00:06:05.460 --> 00:06:07.320
And so I have a pyramid,
so this guy

00:06:07.320 --> 00:06:08.970
gets folded like that.

00:06:08.970 --> 00:06:16.320
And then this guy over
here, can he fold in?

00:06:16.320 --> 00:06:21.150
He's going to fold straight
down and fill that up.

00:06:21.150 --> 00:06:25.360
So this guy, right here, can
fill that right there.

00:06:25.360 --> 00:06:26.800
Let me use some colors.

00:06:26.800 --> 00:06:28.910
And then if this guy,
when I fold it,

00:06:28.910 --> 00:06:30.870
will fill right there.

00:06:30.870 --> 00:06:34.600
And then when I-- and
then this flap

00:06:34.600 --> 00:06:36.810
won't fill in the blank.

00:06:36.810 --> 00:06:39.040
This flap won't go where
it's supposed to.

00:06:39.040 --> 00:06:42.160
This flap will just re-cover
this flap again.

00:06:42.160 --> 00:06:45.310
So choice A is not the
correct choice.

00:06:45.310 --> 00:06:47.230
Let's see choice B.

00:06:47.230 --> 00:06:49.780
So if I flip this over,
that flap is just

00:06:49.780 --> 00:06:52.430
going to go right there.

00:06:52.430 --> 00:06:54.510
So that flap is going
to go right there.

00:06:54.510 --> 00:06:57.690
And then, let's say, this
flap is going to

00:06:57.690 --> 00:07:01.110
go over right there.

00:07:01.110 --> 00:07:06.400
And then this flap, this flap
right here, let's see, if this

00:07:06.400 --> 00:07:10.170
slap, this side right there is
going to be that side, so this

00:07:10.170 --> 00:07:13.930
flap is going to go right over
on this side, just like that.

00:07:13.930 --> 00:07:18.330
And then this last flap--
remember this edge is that

00:07:18.330 --> 00:07:19.080
edge, right there.

00:07:19.080 --> 00:07:20.770
So this last flap will
just fill in the

00:07:20.770 --> 00:07:21.630
blank, right there.

00:07:21.630 --> 00:07:25.600
So it looks like B works--
will actually form a

00:07:25.600 --> 00:07:27.610
rectangular or a
square pyramid.

00:07:27.610 --> 00:07:31.170
Let's see if these other ones
work, before I rule them out.

00:07:31.170 --> 00:07:35.800
So this flap over here, this
top flap, when you flip it

00:07:35.800 --> 00:07:38.800
over, is going to-- you could
imagine looking at the pyramid

00:07:38.800 --> 00:07:42.780
from the top-- and then this
edge, over here-- let me do it

00:07:42.780 --> 00:07:46.410
in pink-- this edge over here
would flip down and

00:07:46.410 --> 00:07:47.770
fill up that edge.

00:07:47.770 --> 00:07:51.280
But then this edge, right here,
will just, when you fold

00:07:51.280 --> 00:07:54.210
this, will just go on top
of that same edge again.

00:07:54.210 --> 00:07:56.030
And then this over here, will
just go over on top of this

00:07:56.030 --> 00:07:56.660
edge, again.

00:07:56.660 --> 00:07:58.440
So we're only getting 1/2
the pyramid, so that's

00:07:58.440 --> 00:07:59.700
not going to work.

00:07:59.700 --> 00:08:00.160
This one.

00:08:00.160 --> 00:08:02.390
You flip this one over.

00:08:02.390 --> 00:08:04.490
It's going to look like that.

00:08:04.490 --> 00:08:06.550
Imagine looking at the
pyramid from above.

00:08:06.550 --> 00:08:08.820
Then you flip this one over.

00:08:08.820 --> 00:08:12.150
It's going to look-- let me
switch colors-- it's going to

00:08:12.150 --> 00:08:15.100
look like this, this
part right here.

00:08:15.100 --> 00:08:20.390
And then this piece-- let me do
it in purple-- this piece

00:08:20.390 --> 00:08:22.730
right here, because this edge
is going to be this edge, is

00:08:22.730 --> 00:08:24.165
going to fill that
right there.

00:08:24.165 --> 00:08:25.900
But that's exactly where
this piece would go

00:08:25.900 --> 00:08:26.500
when you flip it.

00:08:26.500 --> 00:08:28.760
So we're going to have an empty
space right there, so

00:08:28.760 --> 00:08:29.820
that also doesn't work.

00:08:29.820 --> 00:08:32.400
So B is our answer.

00:08:32.400 --> 00:08:35.659
Next problem.

00:08:35.659 --> 00:08:38.789
Use the figure below to answer
the question that follows.

00:08:38.789 --> 00:08:40.450
OK, they give us a
little triangle.

00:08:40.450 --> 00:08:43.070
If equilateral triangle ABC--
well, equilateral tells us

00:08:43.070 --> 00:08:47.080
that all of the slides are
equal-- represents one of two

00:08:47.080 --> 00:08:50.280
congruent halves-- congruent
means completely sides and

00:08:50.280 --> 00:08:53.030
angles are the same-- congruent
halves of a figure

00:08:53.030 --> 00:08:55.880
that has AB as a line
of symmetry.

00:08:55.880 --> 00:08:57.425
So AB is a line of symmetry.

00:08:57.425 --> 00:08:59.160
And this is 1/2 the figure.

00:08:59.160 --> 00:09:02.090
So the rest of the figure is
going to look like this.

00:09:02.090 --> 00:09:04.570
So the line of symmetry, so
you'd literally just-- the

00:09:04.570 --> 00:09:06.992
line of symmetry is going
to look like that.

00:09:06.992 --> 00:09:08.990
It's going to look like that,
and that length is equal to

00:09:08.990 --> 00:09:10.360
that length, is equal
that length, is

00:09:10.360 --> 00:09:11.170
equal to that length.

00:09:11.170 --> 00:09:13.730
So all four sides
are the same.

00:09:13.730 --> 00:09:15.750
Then the entire figure
is, well, all four

00:09:15.750 --> 00:09:16.640
sides are the same.

00:09:16.640 --> 00:09:18.680
It's a little bit skewed,
so it can't be a square.

00:09:18.680 --> 00:09:22.790
It's going to be a rhombus.

00:09:25.410 --> 00:09:27.420
Problem 40.

00:09:27.420 --> 00:09:30.980
Use the diagram below to answer
the question-- well, we

00:09:30.980 --> 00:09:32.110
know how to do that.

00:09:32.110 --> 00:09:34.330
Three straight lines
intersect to form a

00:09:34.330 --> 00:09:36.220
triangle, as shown above.

00:09:36.220 --> 00:09:39.080
What is the measure
of angle x?

00:09:39.080 --> 00:09:41.180
Well here, we just use a couple
of our angle rules.

00:09:41.180 --> 00:09:42.900
If you watched any of my videos
in the past, it's what

00:09:42.900 --> 00:09:44.840
I call the angle game.

00:09:44.840 --> 00:09:46.370
Let's play the angle game.

00:09:46.370 --> 00:09:47.710
We have 60 degrees here.

00:09:47.710 --> 00:09:48.940
This is opposite to the 60.

00:09:48.940 --> 00:09:51.450
It's going to be the same angle,
so that is also going

00:09:51.450 --> 00:09:54.970
to be 60 degrees, right there.

00:09:54.970 --> 00:09:56.950
This is 115 degrees.

00:09:56.950 --> 00:09:59.570
It is going to be supplementary
to this angle,

00:09:59.570 --> 00:10:01.880
right here, which means
that this angle plus

00:10:01.880 --> 00:10:04.130
115 is equal to 180.

00:10:04.130 --> 00:10:10.270
So if we call this x, x plus
115 is equal to 180.

00:10:10.270 --> 00:10:13.870
Subtract 115 from both sides.
x is equal to--

00:10:13.870 --> 00:10:16.230
80 minus 15 is what?

00:10:16.230 --> 00:10:18.880
80 minus 10 is 70.

00:10:18.880 --> 00:10:24.030
70 minus 5 is 65.

00:10:24.030 --> 00:10:27.910
So this angle right here
is 65 degrees.

00:10:27.910 --> 00:10:32.190
And this angle-- let's call that
y-- y plus 65 plus 60.

00:10:32.190 --> 00:10:34.080
And I don't want you to confuse
this x with that x.

00:10:34.080 --> 00:10:37.440
I probably should've called that
some other number, but

00:10:37.440 --> 00:10:41.190
I'm just saying 65 plus 175 is
going to be equal to 180.

00:10:41.190 --> 00:10:42.650
That's how I got the 65.

00:10:42.650 --> 00:10:44.190
They're supplementary.

00:10:44.190 --> 00:10:48.020
And now let's figure out what
this angle y over here is.

00:10:48.020 --> 00:10:55.460
y plus 60, plus 65 has to be
equal to 180 because they form

00:10:55.460 --> 00:10:55.990
a triangle.

00:10:55.990 --> 00:10:58.170
The angles in a triangle
add up to 180.

00:10:58.170 --> 00:11:00.520
So y plus-- add these
two together--

00:11:00.520 --> 00:11:03.740
125 is equal to 180.

00:11:03.740 --> 00:11:07.840
y is equal to 180 minus 125.

00:11:07.840 --> 00:11:10.010
80 minus 25 is what?

00:11:10.010 --> 00:11:11.180
55.

00:11:11.180 --> 00:11:15.540
So y is equal to 55 degrees.

00:11:15.540 --> 00:11:17.140
We're not quite done.

00:11:17.140 --> 00:11:19.860
y is equal to 55 degrees.

00:11:19.860 --> 00:11:22.210
Now, x is supplementary to y.

00:11:22.210 --> 00:11:27.980
So you say x plus 55 is
equal to 180 degrees.

00:11:27.980 --> 00:11:31.340
Well, we already know what
plus 55 is equal to 180.

00:11:31.340 --> 00:11:32.320
125.

00:11:32.320 --> 00:11:37.000
So we say x is equal to 180
minus 55, which is equal to

00:11:37.000 --> 00:11:43.910
125 degrees, which is choice
C, right there.