Problem 89 - Introductory Algebra for College Students

Question

What happens to the volume of a sphere if its radius is doubled?

Step-by-Step Solution

Answer

The volume of the sphere increases by a factor of 8 when the radius is doubled.

Step 1: Identifying the given and required values

The given value in this problem is that the radius of a sphere is doubled. The required value is how this action affects the volume of the sphere. The initial radius is \(r\) and the doubled radius is \(2r\). The volume with the initial radius is \(V = \frac{4}{3} \pi r^3\). We want to find the new volume when the radius is doubled.

Step 2: Calculate the volume with the doubled radius

We will now substitute \(2r\) in the place of \(r\) in the volume formula. This yields \(V' = \frac{4}{3} \pi (2r)^3\) . This multiplies out to \(V' = \frac{4}{3} \pi 8r^3\), which simplifies to \(V' = 8 \times \frac{4}{3} \pi r^3\).

Step 3: Compare the two volumes

We compare the volume of the sphere with radius \(r\) and the volume of the sphere with radius \(2r\). This comparison \(V' / V = 8 \times \frac{4}{3} \pi r^3 / \frac{4}{3} \pi r^3\) simplifies to \(V' / V = 8\). This means the volume of the sphere is multiplied by 8 when its radius is doubled.