Question

a. what happens to the surface area of a sphere if the radius is doubled?
b. what happens to the surface area of a sphere if the radius is tripled?

a. the surface area is multiplied by

b. the surface area is multiplied by

Explanation:

Step1: Recall the formula for the surface area of a sphere

The surface area formula of a sphere is \(S = 4\pi r^{2}\).

Step2: Analyze when the radius is doubled (\(r

ightarrow2r\))
Substitute \(r = 2r\) into the formula: \(S'=4\pi(2r)^{2}=4\pi\times4r^{2} = 4\times(4\pi r^{2})\). Since \(S = 4\pi r^{2}\), then \(S' = 4S\).

Step3: Analyze when the radius is tripled (\(r

ightarrow3r\))
Substitute \(r = 3r\) into the formula: \(S''=4\pi(3r)^{2}=4\pi\times9r^{2}=9\times(4\pi r^{2})\). Since \(S = 4\pi r^{2}\), then \(S'' = 9S\).

Answer:

a. \(4\)
b. \(9\)