Question

A company manufactures cylindrical paint cans with open tops with a volume of 26,000 cubic centimeters. What should be the dimensions of the cans in order to use the least amount of metal in their production? (Round your answer to two decimal places.)

radius _______ cm
height _______ cm

Answer 1

1) Volume, V = Area of the base * heigth = (πr^2)*h = 26,000 cm^3

h = 26,000 / (πr^2)

2) Dimesions of the cans

A = Lateral side + base

Lateral side = 2πrh
Base = π(r^2)

A =  2πrh + πr^2

A = 2πr[26,000/πr^2] + π(r^2) = 52,000/r + π(r^2)

3) Optimization

A' =0

A' = -52,000/r^2 + 2πr =0

-52,000 + 2πr^3 = 0

r^3 = 26,000/π

r = ∛[26,000/π]

r = 20.22 cm
h = 26,000 / [π(20.22)^2] = 20.24  cm