Understanding the Problem: Cone and Sphere Volumes

This problem involves a right circular cone and a sphere with equal volumes. We are given a relationship between the radius and altitude (height) of the cone and asked to find the ratio of the cone's radius to the sphere's radius.

Setting Up the Volumes

First, let's write down the formulas for the volume of a cone and a sphere:

• Volume of a right circular cone, \(V_{cone} = \frac{1}{3} \pi r_c^2 h_c\), where \(r_c\) is the radius of the base and \(h_c\) is the altitude (height).
• Volume of a sphere, \(V_{sphere} = \frac{4}{3} \pi r_s^3\), where \(r_s\) is the radius of the sphere.

Using the Given Relation

The problem states that the radius of the base of the cone is the same as one-third of its altitude. We can write this relation as:

\(r_c = \frac{1}{3} h_c\)

From this, we can express the altitude of the cone in terms of its radius:

\(h_c = 3r_c\)

Equating the Volumes

The problem also states that the volume of the cone and the volume of the sphere are equal:

\(V_{cone} = V_{sphere}\)

Substitute the volume formulas into this equation:

\(\frac{1}{3} \pi r_c^2 h_c = \frac{4}{3} \pi r_s^3\)

Substituting the Altitude Relation

Now, substitute the expression for \(h_c\) (\(h_c = 3r_c\)) into the volume equation:

\(\frac{1}{3} \pi r_c^2 (3r_c) = \frac{4}{3} \pi r_s^3\)

Simplifying the Equation

Let's simplify the left side of the equation:

\(\frac{1}{3} \pi r_c^2 (3r_c) = \pi r_c^3\)

So the equation becomes:

\(\pi r_c^3 = \frac{4}{3} \pi r_s^3\)

Finding the Ratio of Radii

We want to find the ratio of the radius of the cone to the radius of the sphere, which is \(\frac{r_c}{r_s}\). Let's rearrange the equation to isolate the terms involving \(r_c\) and \(r_s\):

Divide both sides by \(\pi\) (since \(\pi \neq 0\)):

\(r_c^3 = \frac{4}{3} r_s^3\)

Divide both sides by \(r_s^3\) (assuming \(r_s \neq 0\)):

\(\frac{r_c^3}{r_s^3} = \frac{4}{3}\)

This can be written as:

\(\left(\frac{r_c}{r_s}\right)^3 = \frac{4}{3}\)

To find \(\frac{r_c}{r_s}\), take the cube root of both sides:

\(\frac{r_c}{r_s} = \sqrt[3]{\frac{4}{3}}\)

\(\frac{r_c}{r_s} = \frac{\sqrt[3]{4}}{\sqrt[3]{3}}\)

So, the ratio of the radius of the cone to the radius of the sphere is \(\sqrt[3]{4} : \sqrt[3]{3}\).

Conclusion on Cone and Sphere Ratio

Based on the calculations, the ratio of the radius of the cone to the radius of the sphere is \(\sqrt[3]{4} : \sqrt[3]{3}\).

Revision Table: Geometry Volume Formulas

Additional Information: Properties of Cones and Spheres

Let's look a bit deeper into the shapes involved in this problem: the cone and the sphere.

Right Circular Cone

• A right circular cone is a three-dimensional geometric shape that tapers smoothly from a flat base (which is a circle) to a point called the apex or vertex.
• The axis of the cone, which connects the apex to the center of the base, is perpendicular to the base.
• The distance from the apex to any point on the circumference of the base is called the slant height.
• The altitude or height is the perpendicular distance from the apex to the base.

Sphere

• A sphere is a perfectly round geometrical object in three-dimensional space that is the surface of a perfectly round ball.
• It is defined as the set of all points that are equally distant from a given point, which is the center.
• The distance from the center to any point on the surface is the radius (\(r\)).
• All points on the surface are equidistant from the center.

Understanding the basic properties and volume formulas for these shapes is crucial for solving geometry problems like this one, especially those involving equal volumes or surface areas.