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.
b. choose the correct answer below
a. the area of the circle decreases by 96%.
b. the area of the circle increases by 96%.
c. the area of the circle increases by 40%.
d. the area of the circle in not affected

Explanation:

Step1: Recall the formula for the area of a circle

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

Step2: Analyze part (a)

If the diameter \(d\) is quadrupled (\(d_{new}=4d\)), then the new radius \(r_{new}=\frac{d_{new}}{2}=4r\).
The new area \(A_{new}=\pi r_{new}^{2}=\pi(4r)^{2}=16\pi r^{2}\).
Since the original area \(A = \pi r^{2}\), the new area is \(16\) times the original area.

Step3: Analyze part (b)

If the radius is increased by \(40\%\), the new radius \(r_{new}=(1 + 0.4)r=1.4r\).
The new area \(A_{new}=\pi r_{new}^{2}=\pi(1.4r)^{2}=1.96\pi r^{2}\).
The percentage increase is \(\frac{A_{new}-A}{A}\times100\%=\frac{1.96\pi r^{2}-\pi r^{2}}{\pi r^{2}}\times 100\%=(1.96 - 1)\times100\% = 96\%\)

Answer:

a. A. The area of the circle is 16 times larger.
b. B. The area of the circle increases by 96%.