Suppose a large soup company wants to minimize the surface area of a can that is to contain 64 cubic inches. What should be the dimensions of the can?

Question

phi · Answer

This is a calculus problem?
First, we assume a can is a cylinder  (otherwise, we would choose a sphere, which minimizes surface area per volume)

So, what are the formulas for 
1)  surface area of a cylinder  
2) volume of a cylinder
(google it if you don't know)

anonymous · Answer

yes it's calc.
1) SA= [2\pi r ^{2}+2\pi rh\]
2) V=[\pi r ^{2}h\]

phi · Answer

use eq 2 with V= 64 and solve for h in terms of r
substitute this expression for h into the first equation.
you now have an expression for surface area as a function of r
take the derivative wrt to r, set to 0, and solve for r
then find h from the 2nd equation

anonymous · Answer

how do I solve it when I have a negative exponent?

phi · Answer

I think the equation is
$$ A= 2 \pi r^2 +128 r^{-1} $$

Are you asking how to take  the derivative?  use the power rule
$$\frac{d }{dx}x^n= n x^{n-1}$$

anonymous · Answer

I can do the derivative to solve for the min, but when I do that I get:
$$4\pi r-128r ^{-2}=0$$
How do I solve for r?

phi · Answer

multiply both sides by r^2

anonymous · Answer

mkay. thanks

phi · Answer

you should get
$$ 4\pi r^3 -128= 0$$
which gives 
$$ r= (\frac{32}{\pi} )^{\frac{1}{3}}$$

anonymous · Answer

yupp that's what I got. Then I just plug it back into where I solved for h in terms of r and get h

phi · Answer

yes.  I got r= 2.1677 and h= 4.3354  (to 4 decimals)

anonymous · Answer

I got the same thing. Thanks for the help :) much appreciated