Optimization(?) Question

Question

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 18 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

Answer 1

Volume of the solid is the volume of the cylinder and hemispheres:

V = πr²h + (4/3)πr³

It's given that V is 18, so:

18 = πr²h + (4/3)πr³ and solve for h (we will use this later):

h = 18/(πr²) - (4/3)r

Total surface area is lateral surface area of cylinder plus surface area of the hemispheres:

A = 36/r + (4/3)πr²h

Substitute h into A:

A = 36/r + (4/3)πr²

Take the derivative of A with respect to r and set it equal to zero:

A'(r) = -36/r² + (8/3)πr = 0

r = 3/cuberoot(2π) centimeters

Hope this was helpful.