Question

given the two concentric circles with center h below, fh = 14. find the area of the shaded region. round your answer to the nearest tenth if necessary.

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: Identify the radii of the two circles

The radius of the larger circle (\( R \)) is \( FH = 14 \), and the radius of the smaller circle (\( r \)) is \( GH = 10 \).

Step3: Calculate the area of the larger circle

Using the formula for the area of a circle, the area of the larger circle (\( A_{large} \)) is \( \pi R^2 = \pi (14)^2 = 196\pi \).

Step4: Calculate the area of the smaller circle

The area of the smaller circle (\( A_{small} \)) is \( \pi r^2 = \pi (10)^2 = 100\pi \).

Step5: Calculate the area of the shaded region (annulus)

The area of the shaded region (annulus) is the difference between the area of the larger circle and the area of the smaller circle. So, \( A_{shaded} = A_{large} - A_{small} = 196\pi - 100\pi = 96\pi \).

Step6: Approximate the value

Using \( \pi \approx 3.14159 \), we get \( 96\pi \approx 96 \times 3.14159 \approx 301.6 \).

Answer:

The area of the shaded region is approximately \( 301.6 \).