Applications of Maxima and Minima Problems

Question

Find the volume of the largest right circular cylinder of maximum volume that can be cut from a circular cone radius 8 cm and a height of 12 cm.

Answer 1

Let x is radius of cylinder. Then its height h = 12 - 3x/2. So, volume V = πx²(12 - 3x/2) = π(12x² - 3x³/2).

V' = π(24x - 9x²/2) = 0; x = 0 or x = 16/3

V'' = π(24 - 9x); V''(0) = 24π > 0. So, at x = 0 we have min

V''(16/3) = - 24π < 0. So, if x = 16/3 cm we have maximum volume max V = π(16/3)²(12 - 3/2·16/3) =

1024π/9 cm³