WHOOO!
Mr. Tarrou.
Wearing a pink shirt again.
Actually this is the same day with many videos.
If you have been watching a bunch of videos.
We are going to model, do some more modeling with functions.
You might notice that there is a part 1 and part 2 because word problems are tough so
I am trying to show you as many examples as I can to help you with whatever class you
guys are in.
So these modeling questions in my PreCalculus book give you a teaser or start to help you
set up learning how to do your related rates problems in Calculus.
We are not doing related rates, but they are similar in setup with a lot of these questions.
So just so you know there is a purpose to all of this.
What I would like to talk about now is volume versus surface area.
We are going to do that with a cube.
I am going to draw a cube here without taking up too much of my precious time.
Precious because I want to do more of these videos today.
And that corner should be up more, but that will do for now.
Ok, so we have a cube and I want to discuss or relate the surface area of this cube to
its volume.
So I want the volume of this rectangular solid to be set at 900 cubic units.
The dimensions are x and x, and have the y value for the height.
So the base is square and the height is adjustable or a different variable, excuse me.
I want to talk about the surface area of this cube, and ultimately I just want it in terms
of x.
So that is the question.
Write me a surface area formula in terms of x.
Well try and just talk about surface area.
If you remember it, there is a formula for the surface area of a cube (this is a box).
But if you don't, let's just figure it out.
Surface area, and this is..
I started to draw this open, but I want to have it closed.
There is a top to this.
Surface area is equal to what?
Well there is a top and bottom.
The bottom and top are x times x for the area of the square on the top and bottom of the
box.
So it is 2x^2 for this particular box.
Length times width for the rectangle of the bottom.
Now this square box has four sides to it, and each side is going to have a width of
x all along the four sides and a height of y.
So let's take this side here because the y is right here.
The area of that rectangle is x times y and there are four of those rectangles going around
the outside of the rectangular solid.
So it is 4xy.
Good!
So this is surface area for any box, any rectangular solid that has a square base.
Well, yeah.
But this is not in terms of x.
I have a y.
So!
Now what?
I am going to relate the surface area of this box to the volume with is 900 un^3.
How do you find the volume of a cube or a box?
It is length times width times height.
So for this particular example it would be x^2 times y.
Well, I don't want the volume to be just anything.
I want the volume to be 900 cubic unit.
Alright, good.
So for this rectangular solid, box, cube...whatever you want to call it.
It is not a cube by the way.
All three sides would have to be the same if it were a cube.
So for this rectangular solid, we have a fixed volume that we must meet.
But the surface area, the shape of the box is fluid.
I just want to make sure that it holds 900 cubic units.
I can use that fixed volume formula to help me get rid of y in my surface area formula.
But to get rid of y, I want this equation solved for y which will then just have x's
on the other side.
So this is going to be y is equal to 900 divided by x squared.
Aha.
So I don't want the surface of any normal rectangular solid with a square base, I want
that rectangular solid to have a volume of 900.
That allows me to take out y and in its place plug in 900/x^2.
That is going in for y.
So the surface area in terms of x is equal to 2x^2 plus 4x times 900/x^2.
We are almost done.
2x^2 plus 3600... not 3000... 3600 over...
This is x on top and this is x squared on the bottom.
Some of those will cancel out leaving you with just an x in the bottom.
That is surface area in terms of x.
BAM!
Alright.
That was a quite bam.
It is kind of a hard question though, wasn't it.
BAM!!!!
There we go:)
Celebrate some more.
Surface area of a box whose volume has to be 900 in terms of x.
Ok, next and last example for this modeling with functions lesson which is now three videos.
We are going to talk about a cone.
This might not even be in your PreCalc book.
If it is, it might just be one near the end.
This is really common or familiar set up in Calculus.
But, you might see it just in there to kind of tease what you are going to see next year.
So I have got a cone.
I am going to fill it with water.
In here is going to be some blue water.
I am shaking the cone around.
And this cone has a radius of 5 and a depth of 15 units.
And then somewhere in there is going to be a circle level of water.
In Calculus that water will be filling and you will be talking about how quickly the
radius or the volume will be changing in a certain time.
We are not in Calculus, so we are just going to talk about...
Or I want to write the volume of the water.
Let me use blue to match the water.
Write a formula for the volume in this cone that is only in terms of the height or the
depth... excuse me...
What am I doing?
Let's write it in terms of the radius.
I can measure how wide the water is on top and then come up with the volume.
Well, the volume of a cone is 1/3 pi r squared times h.
Yes.
Ok.
Spacing out there a little.
Sorry.
ok..
I want to write this formula for volume.. and again it is going to be specific to this
shape of cone... so that it is in terms of r and not r and h.
Ok.
Unlike all of the other problems I have given you where I said
what the area was, or I talked about perimeter, or like the last one where I said the volume
has to be this and then I talked about surface area, I am not giving you a design where I
am giving you two pieces of information that have formulas attached to them.
So somehow I have to work out of this diagram how I could get the h out and replace it with
something that just has r in it.
hmm.
Well, let's see.
Cones have...
They are straight on the side, yes.
So straight line leaning over to the right.
Isn't that kind of like slope.
If I go from here up to the top and over, I am looking at a slope of the side of this
cone.
I am looking at a slope of 3, up 15 and over 5.
Well 15/5 reduces down to 3 over 1.
So if I go up from the bottom up to the water level, well I don't know how tall that is
but I have some r value here.
And we are going to have this h value.
How does h relate to r? h is going to be equal to 3 times the radius.
hmm...
So whatever, I go up 15 and over 5... so my slope is rise over run.
That means that my height is going to be 3 times whatever my radius is.
Oh.
So h=3r.
I can take out h and put in 3r in its place.
My formula becomes 1/3 pi... oh actually no.
1/3 times 3 is 1.
Excuse me.
So pi r cubed.
So for a cone, specific cone where the depth is three times the radius, my volume of my
cone in terms of r is just pi*r^3.
Pretty cool.
Now for this cone let's say I wanted to measure the volume of water just by the depth.
Like maybe i am going to put a stick down to the bottom and measure how much of the
stick is wet.
So maybe I want to measure the volume in terms of the depth of the water, or the height of
the water.
Let me use a different color so you realize I am doing a different example.
I want the volume in terms of h now.
How can I do that?
Well, same cone and depth is still 3 times the radius.
Well, if I want to take out the radius and talk about the volume in terms of just h instead
of r, I can take my relationship between the height and the radius I just noticed...
I can solve it for r because now I want the r out and want just h's in there.
That is going to be h/3 equals r.
OH!
So I can take out the r, replace it with h/3, and remember that it was squared.
We get 1/3*pi*h^2/9*h and get 1/27 pi... and h squared times h is h cubed.
That is volume in terms of h.
BAM!
Alright, thank you very much:)