- #1

toforfiltum

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## Homework Statement

A cylindrical metal can is to be manufactured from a fixed amount of sheet metal. Use the method of Lagrange multipliers to determine the ratio between the dimensions of the can with the largest capacity.

## Homework Equations

## The Attempt at a Solution

$$V(r,h)=\pi r^2h$$

$$2 \pi rh=k$$

Let $$g(r,h)=2 \pi rh$$

$$\nabla V(r,h)=[2 \pi rh, \pi r^2]$$

$$\nabla g(r,h)=[2 \pi h, 2 \pi r]$$

$$2 \pi rh = c(2 \pi h)$$

$$\pi r^2=c(2 \pi r)$$

$$2 \pi rh=k $$

Now, here I'm stuck. If I eliminate ##c##, I get a nonsensical equation such as ##r= \frac{1}{2}r##.

So, I tried another way. Since ##2 \pi rh=k##, I let ##k- 2 \pi ch##, which after some algebraic manipulation gives me the answer ##h=\frac{k}{\pi r}##

But my answer is wrong. The answer is that the height of cylinder equals its diameter. I don't know how to come to that conclusion.

Is my method wrong?

Thanks.