Question

how many times larger is the volume of a sphere when its radius is doubled?
a. 8 times
b. 6 times
c. 2 times
d. 4 times

a spheres radius is tripled. how does its volume change?
a. the volume increases by six times.
b. the volume doubles.
c. the volume triples.
d. the volume increases by 27 times.

Explanation:

Step1: Recall the volume formula of a sphere

The volume formula of a sphere is \(V = \frac{4}{3}\pi r^{3}\).

Step2: Calculate the volume when the radius is doubled

Let the original radius be \(r\), then the original volume \(V_1=\frac{4}{3}\pi r^{3}\). When the radius is doubled (\(r_2 = 2r\)), the new volume \(V_2=\frac{4}{3}\pi(2r)^{3}=\frac{4}{3}\pi\times8r^{3}=8\times\frac{4}{3}\pi r^{3}=8V_1\).

Step3: Calculate the volume when the radius is tripled

Let the original radius be \(r\), original volume \(V_1=\frac{4}{3}\pi r^{3}\). When the radius is tripled (\(r_3 = 3r\)), the new volume \(V_3=\frac{4}{3}\pi(3r)^{3}=\frac{4}{3}\pi\times27r^{3}=27\times\frac{4}{3}\pi r^{3}=27V_1\).

Answer:

For the first question: A. 8 times.
For the second question: D. The volume increases by 27 times.