Last week we learned about all of the
different ways to solve a quadratic
equation, this week we will focus on some of the applications of the quadratic equation.
These all have to deal with
different applications related to
geometry, so the first is that we have a
space in our wall where we wish to put a fish tank.
The width must be 18 inches, the
length is twice the height decreased by
two, so what we have is we have a three
dimensional tank. They've told us that
the width is eighteen inches we don't know
the height and we don't know the length
so I'm going to let "h" equal to
height and since the lengh is twice the
height decreased by 2 I'm going to call
the length 2h - 2. So if the height was "h"
then I would double it and then decrease it by two to find the length. We are
talking about total volume here and we
know that the volume would be length
times width times height so now we're
just going to plug in the values that we
know. I know that the volume is 7560
cubic inches. I know that the length is
2h - 2, the width is 18 and the height
is "h" , so here's my equation now
notice that i have used only one variable
so I would not want to come in and use
"h" for height and "l" for for  length because
there's no way to solve an equation with
two variables unless you have two
equations in which case you could find
where they cross which is when you solve
a system of equations.  In this case
though notice I've used just one variable
and I've set my second unknown in
relationship to the unknown that I had
to begin with, so now it's just a matter
of doing the math and going to multiply all
of this out so I'm going to take
18 x h and then distribute
it to both terms, so 18 h times 2 h
is 36 h squared and then 18 h
x -2 is -36 h.
I ofcourse want everything on
the same side of the equal sign so I'm
going to subtract the 7560 over to the
other sides and I don't like, it's a minus.
I don't like the fact that I've got these
very very large numbers to deal with
because that will make it hard to factor
and even hard to use the quadratic formula
so I'm going to factor 36 out of
everything and that gives me
h squared minus h and minus 210 qual to zero.That makes it much easier now I'm looking for
numbers that multiply to 210 that 
have a difference of one and that turns
out to be 14 and 15 and so it's h minus 15. h plus 14 because they had
different signs and the larger number
was negative which means setting each of
these equal to 0 , I would find that h is 
positive 15 then h is  negative 14. Now ofcourse
this is a story problems so it
does not make sense to have a negative
answer so I know that h is 15. However, 
this asks for the dimensions of the tank.
The dimensions of the tank would be
h which is 15 inches by 18 inches by
2h-2 so if h is 15 that's two times 15
which is 30 - 2 or 28 inches. So those are
the dimensions of the tank.
Use the same example, we know that the
volume of the tank is 7560 cubic inches
and one gallon of water which is the
same as 231 cubic inches weighs
8.345 pounds were trying to find
how was the weight of the water in our tank.
So it makes sense that I'm going to
turn this value my volume into gallons
so that i can multiply that by how much
a gallon weighs, so i'm going to take 7560 cubic
inches divided by well  i'll just say over 1 and then i'll say times 231 cubic inches to 1 gallon now notice
what will happen is that the inches cube
will cancel and my answer will be in gallons.
Dividing that or multiplying
and dividing gives me an answer of
32.727  except for repeating typically I will
leave that value in my calculator now I
have gallons and I know that each of
those weighs 8.345 pounds so that I'm
going to multiply that times 8.345 and that should give me
pounds. I end up with about 273.1 pounds, so that
would be the answer to my question is
way too heavy for me to carry by myself.
We have three sides of a right triangle
so never a bad idea to make yourself a
little picture three sides of a right
triangle are represented by three
consecutive even integers so consecutive
meaning in a row integers meaning
there's no fractions decimals etc. so if
I say that X is the first integer and
they're even integers,  even integers are
two units apart two to four would be two units
apart. Four to six its two units apart so i'm gonna say that X + 2 is my second and two
more than X + 2 which is X + 4 is my third. Now i know at a  right
triangle the longest side is the
hypotenuse so this is X + 4 then it
doesn't matter which one is X or X+2.
Another thing I know about right
triangles is that they use the
Pythagorean theorem which is leg
squared plus leg squared equals
hypotenuse squared. So X squared plus
(X+2) square those were the two legs, 
equals (X+4) square and and thats
the Pythagorean theorem. So now I'm just
going to foil everything out
remember this is X squared,  this is not
X squared + 4, what we do is we take
(X +2)  times (X+2) and foil it
out so that's what that means is take
that twice and then foil it  out  so that would  be  X
squared + 2X + 2X + 4 which would
ofcourse be X squared + 4X + 4, doing the same thing over here
I would get X squared plus 8x + 16 and I
need to get everything to one side so
I'm going to get rid of this X squared.
I'm going to get rid of the 8 X.
I'm gonna get rid of this 16, and on the
right sides I just have zero which is
what I wanted. On the left side I have
X squared, I have -4X
and I have -12 this does happen to
factor into (X-6) (X+2) which
would give me my two solutions of
X = 6 and X = -2 . Again the -2 would not make
sense because we're talking about
lengths and I couldn't  have a length  of -2
so the integers are 6 and then X+2 would be 8 and X+4 would be 10.
So the lengths of the triangle are
6, 8 and 10 they don't specify if they're
inches or feet etc but those are
the integers used for the length of the triangle.
This one's having to do with the
circle so we have a clock face and the
minute hands which of course is the long one when it goes all the way around.
It covers an area of 200 square inches
which is hen telling us that the area of the
circle is 200 square inches.
They're asking what is the length of the
minute hand so the minute hand is the
whole length of the clock from the
center which of course is a radius so
essentially they're saying here's the
area find the radius. We know that the
area of a circle is pi r squared so
that's the formula tha we will use. They give
us that the area is 200, pi we're just going to use 3.14  and radius squared we don't know.
I'll divide each side by 3.14 which
of course will give me r squared all by
itself over here on this side I get
63.69 etc. I'll leave that entire
decimal in my calculator and then I
will take the square root of it and that
gives me rounded to the nearest whole
inch about 8 inches about 8
inches would be the radius a little bit
less than 8.
Here's an ice cream shop he wants a very
large ice cream for his sign but the
county's says make sure that it's not
taller than 15 feet so our job is to
determine if you can have this ice cream
cone based on all of the information
they've given us in the question. So this
sign needs to be big
the height of the bottom triangle is
three times the height of the
semi-circle on top so if this distance
which is of course the radius would be
X, then this distance would be 3X
because it would be 3 times the
height of the smaller and of course this
distance all the way across would be
2X because it would be 2 radii, so
now we have everything we need to figure
out the area or to figure out the X's
based on the total area of 74 square feet
so first let's just look at the
area of each of these parts of the
figure. If I were to find the area of the
figure this is a semi-circle so half of
a circle, a circle is pi r squared, I'm
going to take half of that. A triangle is
one-half times base times height so
that's how I'm going to find the area and then
now i'm going to plug in the values that I know, area is 74, one-half times 3.14 times r squared
which in our case is X squared
and then one-half times the base which
we said was 2X times the height
which we said is 3X , so those are
all of the values and of course our only
unknown is X. The left side is 74 the
right side i have 1.57 X squared + 3X squared notice there's no X to the
first value so in this case I will not
move the 74 over I'm going to
leave the 74 on the left I'll combine
like terms over here so 4.57 X squared.
I'll divide both sides by 4.57
that leaves me with X squared equals
about 16.19 again I'll leave that decimal
on my calculator take the square
root of each side and get the X is
about 4.02. I'm going to round that answer
to the nearest foot so I'm going to say
it's about 4 feet but the question
says can he put up his cone if X is 4.
So,  if X is 4 then that means this
distances is 4 and this distance is 12
leaving us with a total height of 4 + 12  or 16 feet they won't let us
have anything taller than 15 feet so no
sorry Morris you can't have your sign
This last question is one for you to do
on your own this is one very similar to
our last question so press pause try
this question and then press play to see
how you did. Here we have basically a
kite thats in a diamond shape it tells us that
the height of this smaller triangle is
3 feet it tells us that the height
of the lower triangle is 2 feet
shorter than the length of the shared
base so let's call this shared base X
all the way across is X, the height of
this triangle is 2 feet shorter than
that so this is going to be X-2
in the area of this figure i'd have to
find the area of two different triangles
so the area which they've told us
is 36 square feet is equal to one-half
base times height of the smaller
triangle, so the base is X the height
is 3. The area of the larger triangle is
one-half base times height of the larger
so the base of the larger is X and the
height of the larger is X-2
From  here we're just going to do a lot
of multiplication. I see that I've got a
lot of one-half's here  i dont love dealing with fractions
so a little trick on clearing fractions
is to take each term by two, so by doing
that on the left side that just gives me
72 on the right side that leaves me with
the one half and the two going away
giving me X times 3 or 3X
and then on this one of course the one half and the two going away
giving me X squared - 2X
so much much easier than dealing with
the one halfs all the way through the
question this one does have an X squared
and an X term so everything's going to go
to one side meaning I'm onna subtract
seventy two from each side and do some
rearranging
i have X squared, I have +X  because those
two combined and then I have -72.
Again I'm always going to
factor if I can, in this case I can
X+9 and X-8 , those two
values multiplied a -72 with the
difference of 1 which leaves me with
an X of -9 and a +8.
Of course -9 does not
make sense, I'm trying to find the base
and the height of the lower trying so
the base of the lower triangle is 8 feet
because that's the value of X.
The height of the lower triangle is X-2 or
8-2 which is 6.
