Question

a. what happens to the area of a circle if its diameter is quadrupled?
b. what happens to the area of a circle if its radius is increased by 40%?
c. what happens to the area of a circle if its circumference is quintupled?
a. choose the correct answer below.
a. the area of the circle is 16 times larger.
b. the area of the circle is 4 times larger.
c. the area of the circle is 16 times smaller.
d. the area of the circle is not affected.

Explanation:

Step1: Recall the formula for the area of a circle

The area of a circle is \(A = \pi r^{2}\), and the diameter \(d = 2r\), so \(r=\frac{d}{2}\), and \(A=\pi(\frac{d}{2})^{2}=\frac{\pi d^{2}}{4}\)

Step2: Analyze the effect of quadrupling the diameter

Let the original diameter be \(d_{1}\), and the new diameter \(d_{2}=4d_{1}\). The original area \(A_{1}=\frac{\pi d_{1}^{2}}{4}\), and the new area \(A_{2}=\frac{\pi d_{2}^{2}}{4}=\frac{\pi(4d_{1})^{2}}{4}=\frac{\pi\times16d_{1}^{2}}{4} = 16\times\frac{\pi d_{1}^{2}}{4}=16A_{1}\)

Answer:

A. The area of the circle is 16 times larger.