A container open at the top is a right circular cylinder having a surface area of 108 cm^3. What should the radius and altitude be in order to provide the largest possible volume?

Answer 1

So the surface area is made up of the circular base and the "sleeve" of the cylinder.
πr^2 + 2πrh = 108
h = (108 - πr^2)/(2πr) = 59/(πr) - r/2

V = π r^2 h
= πr^2(59/(πr) - r/2)
= 59r - (π/2)r^3
dV/dr = 59 - (3π/2)r^2 = 0 for a max of V

(3π/2)r^2 = 59
solve for r, then get h