Find the dimensions of a closed right circular cylindrical can of smallest surface area whose volume is 128pi cm^3

The surface area of a cylindrical can is equal to the sum of the area of two circles and the body of the cylinder: 2πr2 + 2πrh. volume is equal to π r2h.

V = π r2h = 128 pi

r2h = 128

h = 128/r2

A = 2πr2 + 2πrh

A = 2πr2 + 2πr * (128/r2)

A = 2πr2 + 256 π / r

the optimum dimensions is determined by taking the first derivative and equating to zero.

dA = 4 πr - 256 π / r2 = 0

r = 4 cm

h = 8 cm