Question

both circles have the same center. what is the area of the shaded region?
d=103.6 km
38.9 km
write your answer as a whole number or a decimal rounded to the nearest hundredth.
square kilometers
submit
work it out
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Explanation:

Step1: Find radius of inner circle

The diameter of the inner circle \( d = 103.6 \) km, so radius \( r_{inner}=\frac{d}{2}=\frac{103.6}{2} = 51.8 \) km.

Step2: Find radius of outer circle

The width of the ring (shaded region) is 38.9 km, so radius of outer circle \( r_{outer}=r_{inner}+ 38.9=51.8 + 38.9 = 90.7 \) km.

Step3: Calculate area of shaded region (annulus)

The formula for the area of an annulus is \( A=\pi(r_{outer}^2 - r_{inner}^2) \).
Substitute \( r_{outer}=90.7 \) and \( r_{inner}=51.8 \):
\( A=\pi(90.7^2 - 51.8^2)=\pi((90.7 - 51.8)(90.7 + 51.8))=\pi(38.9\times142.5) \)
Calculate \( 38.9\times142.5 = 38.9\times(140 + 2.5)=38.9\times140+38.9\times2.5 = 5446+97.25 = 5543.25 \)
Then \( A=\pi\times5543.25\approx3.1416\times5543.25\approx17411.06 \) (rounded to nearest hundredth)

Answer:

17411.06