Question

the smallest of the three circles with center d has a radius of 8 inches and cb = ba = 4 inches. what is the sum of the areas of all three circles? 80π in.² 96π in.² 208π in.² 464π in.²

Explanation:

Step1: Find radii of three circles

Smallest circle (radius \( r_1 \)): \( r_1 = 8 \) inches.
Middle circle: Radius \( r_2 = 8 + 4 = 12 \) inches (since \( CB = 4 \)).
Largest circle: Radius \( r_3 = 12 + 4 = 16 \) inches (since \( BA = 4 \)).

Step2: Calculate area of each circle

Area formula: \( A = \pi r^2 \).
Smallest area: \( A_1 = \pi(8)^2 = 64\pi \).
Middle area: \( A_2 = \pi(12)^2 = 144\pi \).
Largest area: \( A_3 = \pi(16)^2 = 256\pi \).

Step3: Sum the areas

Total area \( = A_1 + A_2 + A_3 = 64\pi + 144\pi + 256\pi = 464\pi \) square inches.

Answer:

\( 464\pi \) in.² (corresponding to the option: 464π in.²)