Question

Find the maximum volume of right circular cylinder that can be inscribed in a cone of altitude 12 cm and base radius 4 cm, if the axes of the cylinder and con coincide.

Answer 1

Try to make a sketch of a cylinder inside a cone
Draw in the altitude, let the height be h
let the radius of the cylinder be r
Look at a cross section of the diagram.
the altitude from the top of the cylinder to the vertex of the cone is 12-h
and by similar triangles
(12-h)/r = 12/4 = 3/1
3r = 12-h
h = 12-3r

V(cylinder) = πr^2 h
= πr^2 (12-3r)
= 12πr^2 - 3πr^3
dV/dr = 24πr - 9πr^2 = 0 for a max of V
3πr(8 - 3r) = 0
r = 0 , clearly yielding a minimum Volume
or
r = 8/3

max V = ....

(you do the button pushing)