find the dimensions that will minimize the cost of metal to make th can

Question

The Cylindrical can is without a top and is made to contain 500 cm^3 of liquid. the metal used to make the can costs 0.10 $ per square centimeter find the dimensions that will minimize the cost of metal to make the can. 

Answer 1

Minimizing the metal means minimizing the surface area of the can (we will assume the thickness of the metal stays the same no matter what the dimensions are). We don't really need to worry about the cost of $0.10/cm² because it's the same for both bottom and sides so it doesn't really matter.

The surface area of the sides of a can is:

SA_side = 2πrh

The surface are of the bottom of a can is:

SA_bottom = πr²

So the surface area of the entire can is:

SA = 2πrh + πr²

But we are given a restriction that the can is to hold 500 cubic cm of liquid. The volume of a can is:

V = πr²h so

500 = πr²h or h = 500/(πr²)

Substituting 500/(πr²) for "h" into the surface area equation we get:

SA = 2πr(500/(πr²)) + πr²

SA(r) = 1000/r + πr²

To minimize, take the derivative and set it equal to zero:

SA'(r) = -1000/r² + 2πr

-1000/r² + 2πr = 0

1000/r² = 2πr

500 = πr³

r = cuberoot(500/π)

r = 5.42 cm

Since h = 500/(πr²)

h = 500/(π(5.42)²) = 5.42 cm