A manufacturing company wants to minimize the cost of materials for a pop can which needs to hold 640 mL. Find the radius and height (in centimeters) of the cylindrical can with minimum surface area that holds this volume. Give your answer as a decimal number accurate to at least one decimal place.

Question

A manufacturing company wants to minimize the cost of materials for a pop can which needs to hold 640 mL. Find the radius and height (in centimeters) of the cylindrical can with minimum surface area that holds this volume. Give your answer as a decimal number accurate to at least one decimal place.

Thanks.

Answer 1

Assume the cost will be a function of the surface area SA.

radius of can ---- r
height of can ---- h
V = πr^2 h = 640
h = 640/(πr^2)

SA = 2πr^2 + 2πrh
= 2πr^2 +2πr(640/(πr^2)
= 2πr^2 + 1280/r
d(SA)/dr = 4πr - 1280/r^2 = 0 for a min of SA

4πr = 1280/r^2
r^3 = 320/π
r = appr 4.67
then h = 640/(π(4.67..)^2 = appr 9.34

did you notice that height = diameter ?