Posted by **Anh** on Friday, December 9, 2011 at 8:09pm.

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.

- calculus -
**Reiny**, Friday, December 9, 2011 at 11:13pm
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)

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