Question

find the radius and height of a right-circular cylinder of greatest volume which can be inscribed in a right-circular cone with a radius of 8 meters and a height of 12 meters

Answer 1

See the attachment.

h/(8 - r) = 12/8

h = (3/2)(8 - r)

h = 12 - (3/2)r

V(r) = πr²(12 - (3/2)r)

= 12πr² - (3/2)πr³

V'(r) = 24πr - (9/2)πr² = 0

48πr - 9πr² = 0

16 - 3r = 0

3r = 16

r = 16/3

V(16/3) = π(16/3)²(12 - (3/2)(16/3))

= π(256/9)(12 - 8)

= π(256/9)(4)

= 1,024π/9 m²