Question

8 mark for review
circle a has a radius of 3n and circle b has a radius of 129n, where n is a positive constant. the area of circle b is how many times the area of circle a?
a 43
b 86
c 129
d 1,849

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: Calculate the area of circle A

For circle A, the radius \( r_A = 3n \). So, the area \( A_A = \pi (3n)^2 = \pi \times 9n^2 = 9\pi n^2 \).

Step3: Calculate the area of circle B

For circle B, the radius \( r_B = 129n \). So, the area \( A_B = \pi (129n)^2 = \pi \times 16641n^2 = 16641\pi n^2 \).

Step4: Find the ratio of the area of circle B to the area of circle A

To find how many times the area of circle B is the area of circle A, we calculate the ratio \( \frac{A_B}{A_A} \).
Substituting the values of \( A_A \) and \( A_B \) we found:
\( \frac{A_B}{A_A} = \frac{16641\pi n^2}{9\pi n^2} \)
The \( \pi n^2 \) terms cancel out, leaving \( \frac{16641}{9} \).
Calculating \( \frac{16641}{9} = 1849 \).

Answer:

D. 1,849