optimization?

Question

A right circular culinder is inscribed in a cone with height H and base radius r. Find the largest possible volume of such a cylinder

Answer 1

Given a right circular cone and an inscribed cylinder.

h = height of cone
r = base radius of cone
x = radius of cylinder
y = height of cylinder
V = volume cylinder

Maximize volume of cylinder.

V = πx²y
x/r = (h - y)/h
x = r(h - y)/h = (r/h)(h - y)

V = π[(r/h)(h - y)]²y = π(r²/h²)(h²y - 2hy² + y³)

Take the derivative and set equal to zero to find the critical values for y.

dV/dy = π(r²/h²)(h² - 4hy + 3y²) = 0
3y² - 4hy + h² = 0
(3y - h)(y - h) = 0
y = h/3,h

Take the second derivative to find the nature of the critical points.

d²V/dy² = π(r²/h²)(-4h + 6y)

For y = h/3
d²V/dy² = π(r²/h²)(-4h + 6h/3) = π(r²/h²)(-2h) < 0
Implies relative maximum which is what we want.

For y = h
d²V/dy² = π(r²/h²)(-4h + 6h) = π(r²/h²)(2h) > 0
Implies relative minimum which is not what we want.
This solution is rejected.

Plugging in y = h/3

V = π[(r/h)(h - y)]²y = π[(r/h)(h - h/3)]²(h/3) = π[(r/h)(2h/3)]²(h/3)
V = π[(r)(2/3)]²(h/3) = π[4r²/9)]²(h/3) = 4πr²h/27