I have no idea how to approach this problem, if someone knows just how to relate h, r with H,R, that would be extremely helpful and I can workout the rest! Thank you in advance.

Given a right circular cone, you put an upside-down cone inside it so that its vertex is at the
center of the base of the larger cone, and its base is parallel to the base of the larger cone. If
you choose the upside-down cone to have the largest possible volume, what fraction of the
volume of the larger cone does it occupy? (Let H and R be the height and radius of the large
cone, let h and r be the height and radius of the small cone. The formula for the volume of a cone is V= 1/3pi(r^2)h

http://mathhelpforum.com/calculus/232420-maximize-cone-inside-cone.html

From his smaller volume V, take the derivative, set to zero, solve for r (for max volume.

Then determine the ratio

To relate the dimensions of the small cone (h, r) to the dimensions of the large cone (H, R), we need to make use of similar triangles.

Let's break down the problem step by step:

1. Consider the two cones as similar figures. The small cone is a scaled-down version of the large cone.

2. We can set up a proportion to relate the corresponding dimensions of the two cones:
(small cone height) / (large cone height) = (small cone base radius) / (large cone base radius)

Mathematically, this can be expressed as:
h/H = r/R

3. Next, let's express the volume of each cone in terms of the given dimensions:
Volume of small cone = (1/3) * π * r^2 * h
Volume of large cone = (1/3) * π * R^2 * H

4. We want to determine the fraction of the volume of the larger cone occupied by the smaller cone. Therefore, we need to find the ratio of their volumes.

5. Substitute the relation from step 2 into the volume formulas from step 3:
Volume of small cone = (1/3) * π * r^2 * (h/H) * R^2
Volume of large cone = (1/3) * π * R^2 * H

6. Simplify the equation by canceling out the common terms:
Volume of small cone = (1/3) * π * r^2 * R^2
Volume of large cone = (1/3) * π * R^2 * H

7. The ratio of the volume occupied by the small cone to the volume of the large cone is:
V_small / V_large = [(1/3) * π * r^2 * R^2] / [(1/3) * π * R^2 * H]

Simplifying the equation further:
V_small / V_large = (r^2 * R^2) / (R^2 * H)

8. Finally, cancel out the common terms:
V_small / V_large = r^2 / H

So, the fraction of the volume of the larger cone occupied by the smaller cone is r^2 / H.