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?
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.
c. choose the correct answer below.
a. the area of the circle is 25 times larger.
b. the area of the circle is 5 times larger.
c. the area of the circle is 25 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 given by \(A = \pi r^{2}\), where \(r\) is the radius of the circle.

Step2: Analyze part (a)

If the diameter \(d\) is quadrupled (\(d_{new}=4d\)), since \(r=\frac{d}{2}\), the new radius \(r_{new}=\frac{d_{new}}{2}=\frac{4d}{2} = 4r\).
The new area \(A_{new}=\pi(4r)^{2}=16\pi r^{2}\).
The ratio of the new area to the old area is \(\frac{A_{new}}{A}=\frac{16\pi r^{2}}{\pi r^{2}} = 16\). So the area of the circle is 16 times larger.

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 \(A_{new}=\pi(1.4r)^{2}=\pi\times1.96r^{2}\).
The percentage change in area is \(\frac{A_{new}-A}{A}\times100\%=\frac{1.96\pi r^{2}-\pi r^{2}}{\pi r^{2}}\times 100\%=(1.96 - 1)\times100\%=96\%\). So the area of the circle increases by \(96\%\).

Step4: Analyze part (c)

The circumference of a circle is \(C = 2\pi r\). If \(C_{new}=5C\), then \(2\pi r_{new}=5\times(2\pi r)\), so \(r_{new}=5r\).
The new area \(A_{new}=\pi(5r)^{2}=25\pi r^{2}\).
The ratio of the new area to the old area is \(\frac{A_{new}}{A}=\frac{25\pi r^{2}}{\pi r^{2}}=25\). So the area of the circle is 25 times larger.

Answer:

a. The area of the circle is 16 times larger.
b. B. The area of the circle increases by 96%
c. A. The area of the circle is 25 times larger