The diagram shows a sphere and a cone.
Diagram NOT accurately drawn
The cone has height $h \mathrm{~cm}$.
The radius of the base of the cone is 3 times the radius of the sphere.
Given that the volume of the sphere is equal to the volume of the cone, find an expression for the radius of the sphere in terms of $h$. Give your expression in its simplest form.
Answer
\(\boxed{r = \frac{9h}{4}}\)
Step 1 ：Let the radius of the sphere be r and the radius of the cone be 3r.
Step 2 ：The volume of the sphere is given by the formula: \(V_{sphere} = \frac{4}{3} \pi r^3\)
Step 3 ：The volume of the cone is given by the formula: \(V_{cone} = \frac{1}{3} \pi (3r)^2 h\)
Step 4 ：Since the volumes are equal, we can set the two formulas equal to each other: \(\frac{4}{3} \pi r^3 = \frac{1}{3} \pi (3r)^2 h\)
Step 5 ：Cancel out pi and 1/3 from both sides: \(4r^3 = 9r^2 h\)
Step 6 ：Solve for r in terms of h: \(r = \frac{9h}{4}\)
Step 7 ：\(\boxed{r = \frac{9h}{4}}\)
