A cylindrical can, open at the top, is to hold 470 cm^3 of liquid. Find the height and radius that minimize the amount of material needed to manufacture the can.

Question

A cylindrical can, open at the top, is to hold 470 cm^3 of liquid. Find the height and radius that minimize the amount of material needed to manufacture the can. Enter your answer with rational exponents, and use pi to represent π.

radius =

height =

Answer 1

Let r = the radius and h = the height

Volume, V = πr²h = 470

h= 470/(πr²)

Surface Area, S = πr² + 2πrh = πr² + 2πr(470/(πr²)) Since the top is open

S(r) = πr² + 940/r After simplifying

S'(r) = 2πr - 940/r²

Setting S'(r) to zero and solving for r

2πr³=940

r = (470/π)^1/3 inches

h = 470/(π(470/π)^2/3 = (470/π)^1/3 inches

Notice that the radius and height are the same to give the minimum surface area.