Question

A rectangular sheet of perimeter 33 cm and dimensions x cm by y cm is to be rolled into a cylinder. What values of x and y give the largest volume? No idea how to solve this problem o_O

Answer 1

Hmmm an optimization problem. Is this for Calculus? Using derivatives to maximize volume and such?

Or is this for a lower level class, such as algebra?
I'm not too familiar with how to solve this without using calculus methods, I would need to think about it if that's the case.

Answer 2

Yeah, it's a problem from my calculus class.

Answer 3

ok cool c:

Answer 4

I have the answer key but I don't know how they got it.

Answer 5

So we want to MAXIMIZE volume.
(Find a critical point for the volume function which we can identify as the maximum).

And we are given a constraint on the perimeter.

So let's do a couple things to start. Let's draw a picture real quick.

Answer 6

okay, that's as far as I could get.

Answer 7

If we roll it up to make a cylinder, we get something like this.

Answer 8

We know the formula for volume of a cylinder right? I think it's ummmm\[\large V=(\pi r^2)h\]

Answer 9

So we have our height (with the way we chose to roll it up).
We need to find the radius.

Answer 10

y is the circumference of the circle.
It's the distance AROUND the cylinder, right?

\[\large C=2\pi r \qquad \rightarrow \qquad y=2\pi r \qquad \rightarrow \qquad r=\frac{y}{2\pi}\]

Answer 11

okay, i see...

Answer 12

so that will give us y then we substitute y in the 2x + 2y =33 formula to get x?

Answer 13

Yes very good!

We want our volume equation in terms of ONE VARIABLE so it's easier to take a derivative of. Using the Constraint equation will allow us to make that substitution! :)

Answer 14

But how do we get r from ( r = y / 2pi) if I don't know the value of r?

Answer 15

See how the CIRCUMFERENCE (distance around the circle) is our y value? We relate the radius to the circumference.