Question stated in description.

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 side view. Using similar triangles, it is easy to see that if the smaller cone has radius r and height h, then

r = 3/5 (9-h)

or,

h = 5/3 (9-r)

The volume of the small cone is thus

v = π/3 r²h = 5π/9 r²(9-r)

dv/dr = 5π/3 r(6-r)

So, the maximum volume occurs when r=6 and h=5

That means that the maximum volume is 60π

The volume of the large cone is 405π

The smaller cone is 1/3 as tall, with 2/3 the radius, so its volume will be 1/3 * (2/3)^2 = 4/27 the larger volume, or 60π