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**[SOLVED] Minimizing Surface Area**

**1. Homework Statement**

A can is to be manufactured in the shape of a circular cylinder with volume = 50.

Find the dimensions of a can that would minimize the amount of material needed to make the can.

**2. Homework Equations**

V = [tex] \pi r^2 h[/tex]

SA = [tex] 2 \pi r^2 + 2 \pi r h [/tex]

**3. The Attempt at a Solution**

I have never done a problem like this so I am unsure how to do it, but here is my attempt.

With the volume equation I solved for h. [tex] h = \frac{50}{\pi r^2} [/tex]

I plugged this value for h into the Surface area equation. [tex] SA = 2 \pi r^2 + 2 \pi r \frac{50}{\pi r^2} [/tex]

which = [tex] 2 \pi r^2 + \frac{100}{r} [/tex]

I then took the derivative of that and set it equal to 0.

[tex] 0 = 4 \pi r - 100 r^-2 [/tex]

[tex] r = \sqrt[3]{\frac{100}{4 \pi}} [/tex]

r = 1.996

Then I plugged that back into the volume equation to solve for h and got h= 3.99.

Could someone tell if this is right and if not where I went wrong. Thanks.