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 208π in.² 96π in.² 464π in.²

Explanation:

Step1: Find the radii of the other two circles

The radius of the smallest circle \(r_1 = DC=8\) inches.
Since \(CB = 4\) inches, the radius of the middle - sized circle \(r_2=DB=DC + CB=8 + 4=12\) inches.
Since \(BA = 4\) inches, the radius of the largest circle \(r_3=DA=DB + BA=12 + 4 = 16\) inches.

Step2: Use the formula for the area of a circle \(A=\pi r^{2}\)

The area of the smallest circle \(A_1=\pi r_1^{2}=\pi\times8^{2}=64\pi\)
The area of the middle - sized circle \(A_2=\pi r_2^{2}=\pi\times12^{2}=144\pi\)
The area of the largest circle \(A_3=\pi r_3^{2}=\pi\times16^{2}=256\pi\)

Step3: Calculate the sum of the areas

\(A = A_1+A_2+A_3\)
\(A=64\pi+144\pi + 256\pi\)
\(A=(64 + 144+256)\pi\)
\(A = 464\pi\)

Answer:

\(464\pi\space in^{2}\)