Question

both circles have the same center. what is the area of the shaded region?
d=74 mi
34 mi
write your answer as a whole number or a decimal rounded to the nearest hundredth.
square miles

Explanation:

Step1: Find radius of inner circle

The diameter of the inner circle \( d = 74 \) mi, so radius \( r=\frac{d}{2}=\frac{74}{2} = 37 \) mi.

Step2: Find radius of outer circle

The width of the ring (distance between inner and outer circle) is 34 mi, so outer radius \( R=r + 34=37+34 = 71 \) mi.

Step3: Calculate area of shaded region (annulus)

The area of an annulus is \( A=\pi R^{2}-\pi r^{2}=\pi(R^{2}-r^{2}) \).
Substitute \( R = 71 \) and \( r = 37 \):
\( R^{2}-r^{2}=71^{2}-37^{2}=(71 + 37)(71 - 37)=108\times34 = 3672 \)
Then \( A=\pi\times3672\approx3.14159\times3672\approx11530.08 \) (rounded to nearest hundredth)

Answer:

\( 11530.08 \)