We are given that a cone and a sphere have equal volumes and equal radii, and that the height of the cone is 36 cm. We need to find the length of the radius $r$ of both shapes.

Step 1: Write the formulas for the volumes of the cone and the sphere.

• The volume $V_{\text{cone}}$ of a cone is given by: $V_{\text{cone}} = \frac{1}{3} \pi r^2 h$ where $r$ is the radius and $h$ is the height of the cone.

• The volume $V_{\text{sphere}}$ of a sphere is given by: $V_{\text{sphere}} = \frac{4}{3} \pi r^3$ where $r$ is the radius of the sphere.

Step 2: Set the volumes equal to each other.

Since the volumes of the cone and the sphere are equal, we can set the two formulas equal to each other: $\frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^3$ Since $\pi$ appears on both sides, it cancels out: $\frac{1}{3} r^2 h = \frac{4}{3} r^3$ Multiply both sides by 3 to get rid of the denominator: $r^2 h = 4 r^3$ Now, divide both sides by $r^2$ (assuming $r \neq 0$): $h = 4r$

Step 3: Substitute the height of the cone.

We are told that the height of the cone is 36 cm. Substituting this value for $h$: $36 = 4r$ Now, solve for $r$: $r = \frac{36}{4} = 9 \, \text{cm}$

Final Answer:

The radius of both the cone and the sphere is 9 centimeters.

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Would you like further details on any step or have any questions? Here are some related questions:

1. How do you find the volume of a truncated cone?
2. What is the surface area formula for a sphere?
3. Can the volume of a cone be greater than that of a sphere if they have the same radius?
4. What is the relationship between the slant height and radius in a cone?
5. How would the radius change if the height of the cone was doubled?

Tip: Always cancel out common terms like $\pi$ early to simplify equations in geometry problems.