Question

A small cone has a vertex touching the base of a large cone. The base of the small cone is parallel to the base of the larger cone.
a.) Find h and r so that the smaller cone has a maximum volume.
b.) State the maximum volume
c.) explain how you know whether your answer is reasonable or not.

(Larger cone has a radius of 9cm, with a height of 15cm. h is the height of the smaller cone, r is the radius of the smaller cone)

Answer 1

Draw a diagram. Using similar triangles, you can see that if the smaller cone has height h, then its radius
r = 3/5 (15-h).
so, h = 5/3 (9-r)
So, now we know that the small cone has volume
v = π/3 r^2 h = π/3 r^2 * 5/3 (9-r)
so, now just find r where dv/dr=0

If you get stuck, come on back and show what you did.