Optimizayion of a can of apple juice?

Question

A cylindrical can is to hold 500 cm^3 of apple juice the design must take into account that the height must be between 6 and 15 cm, inclusive. How should the can be constructed so that a minimum amount of material will be used in the construction? assume no waste.

V is volume and SA is surface area

so far this is what I have:

15>h>6

v=(pi)r^2h
500=(pi)r^2h
500/(pi)r^2=h

SA=2(pi)r(500/(pi)r^2) + 2(pi)r^2

Now I believe I need to find the derivative and set it to zero then solve for r.

The problem is that I can't quite figure out how I'm supposed to find the derivative if someone could give me a hand understanding the process as I have forgotten and my teacher is busy with the latest stuff.

Answer 1

Here's what you do:

The surface area formula SA=2(pi)r(500/(pi)r^2) + 2(pi)r^2 simplifies to 1000/r + 2(pi)r^2 or 1000r^-1 + 2(pi)r^2

Take the derivative of this, and set it equal to zero.
-1000r^-2 + 4(pi)r = 0

You can solve for "r" by getting the terms on opposite sides of the equation and cross-multiplying.

4(pi)r = 1000 / r^2
4(pi)r^3 = 1000
r^3 = 250/(pi)

This gives that r is the cube root of (250 / pi), which is about 4.3

You can find "h" by plugging into the formula 500/(pi)r^2=h

500 / [ (pi)*4.3^2 ] is approximately 8.6

Since 8.6 is within the constraings (between 6 and 15), you have your answer.