Solve Derivative Problem: Find Max Volume of Cylinder w/100in2 Surface Area

[notastring]

Homework Statement

A company wants to build a cylindrical can with a total surface area of 100 square inches, which includes the bottom and top of the can. Find the dimensions that maximize the volume.

Homework Equations

volume of a cylinder = 2*pi*r^2*h
surface area of cylinder = 2*pi*r^2 + 2*pi*r*h

The Attempt at a Solution

I wrote a huge wall of text detailing how I solved the problem, but when I went to post I had gotten logged out due to how much time I spent on the post. So I'm going to make this short and just explicate my process:

I took the formula for constant surface area 100 inches squared and solved for the height in terms of the radius. Then, I plugged that equation (h in terms of r) into the volume and took the derivative of the volume with respect to the radius.

I set the derivative equal to zero to find the radius that gives the max volume, and I got
r = sqrt(50/(pi*3))

Then I plugged r back into the equation for h in terms of r, and simplifying, I got:
h = sqrt(150/pi) - sqrt(50/(3*pi))

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The answer sheet I have says that I should have gotten H = 2R. Indeed my answer reflects this, but should I have arrived at a general solution H = 2R instead of the exact answers I got? How would I go about solving this generally?

Was my approach fine? Could someone maybe solve it for me in an alternate manner? I would really appreciate iany advice. Thank you.

 

[Simon Bridge]

Homework Statement

A company wants to build a cylindrical can with a total surface area of 100 square inches, which includes the bottom and top of the can. Find the dimensions that maximize the volume.

Homework Equations

volume of a cylinder = 2*pi*r^2*h
surface area of cylinder = 2*pi*r^2 + 2*pi*r*h

The Attempt at a Solution

I wrote a huge wall of text detailing how I solved the problem, but when I went to post I had gotten logged out due to how much time I spent on the post.

There a setting that stops you from being logged out like that ... though I used to write out my posts on a text editor and copy-and-paste it over ;)

So I'm going to make this short and just explicate my process:

I took the formula for constant surface area 100 inches squared and solved for the height in terms of the radius. Then, I plugged that equation (h in terms of r) into the volume and took the derivative of the volume with respect to the radius.

I set the derivative equal to zero to find the radius that gives the max volume, and I got
r = sqrt(50/(pi*3))

Then I plugged r back into the equation for h in terms of r, and simplifying, I got:
h = sqrt(150/pi) - sqrt(50/(3*pi))

-----------------------------------------------------------------------------------------------

The answer sheet I have says that I should have gotten H = 2R. Indeed my answer reflects this, but should I have arrived at a general solution H = 2R instead of the exact answers I got? How would I go about solving this generally?

The question asks for the specific solution - the general relation would maximize the volume for any given surface area.

 

[HallsofIvy]

If surface area $S= 2\pi R H+ 2\pi R^2$ and you want to maximize the volume $V= \pi R^2H$ then $\nabla V= 2\pi RH\vec{i}+ \pi R^2\vec{j}$ must be a multiple of $\nabla S= (2\pi H+ 4\pi R)\vec{i}+ 2\pi R\vec{j}$. That means we must have $2\pi RH= \lambda(2\pi H+ 4\pi R)$ and $\pi R^2= 2\lambda\pi R$ for some constant $\lambda$. We can eliminate $\lambda$ by dividing the first equation by the second:
$$\frac{2\pi RH}{\pi R^2}= \frac{2\pi H+ 4\pi R}{2\pi R}$$
$$\frac{2H}{R}= \frac{H+ 2R}{R}$$
$$2H= H+ 2R$$
$$H= 2R$$

 

[notastring]

Hey thanks!