Question
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?
Answers
Reiny
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
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