Problem 89 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

When the radius of a sphere is doubled, its volume becomes 8 times larger.

Step 1: Understand the problem

You are asked to solve what happens to the volume of a sphere when its radius is doubled. The volume of a sphere can be calculated using the formula \(V = \frac{4}{3}\pi r^3 \).

Step 2: Double the Radius

Let's double the radius, meaning instead of \(r\) we will use \(2r\) in the formula. So the formula will look like this: \(V_{new} = \frac{4}{3}\pi (2r)^3\).

Step 3: Simplify the expression

When you simplify the expression, you'll get \(V_{new} = \frac{4}{3} \pi 8r^3\). Then, you can simplify it further to \(V_{new} = 8(\frac{4}{3}\pi r^3)\). Key point to notice here is that \(\frac{4}{3}\pi r^3\) is the original volume of the sphere.

Step 4: Compare the new volume to the original volume

Here \(V_{new}\) is 8 times the original volume \(V\). So, when the radius of a sphere is doubled, its volume becomes 8 times larger.