Question

item 23
in the figure shown, a squar circumscribed about a circle with an 8 - inch radius. diameter \\( \overline { a b } \\) is parallel to 2 sides of the square, and \\( \overline { c d } \\) is
some regions, bounded by the square, the circle, \\( \overline { a b } \\), and \\( \overline { c d } \\), are shaded. which of the following is closest to the total area, in square inch

a. 64
b. 176
c. 224
d. 256

Explanation:

Step1: Calculate the area of the square

Since the radius of the circle is \(r = 8\) inches, the diameter of the circle \(d=16\) inches. The side length of the square is equal to the diameter of the circle.
The area of the square \(A_{square}=s^{2}\), where \(s = 16\) inches. So \(A_{square}=16\times16 = 256\) square inches.

Step2: Calculate the area of the circle

The area of the circle \(A_{circle}=\pi r^{2}\), with \(r = 8\) inches. So \(A_{circle}=\pi\times8^{2}=64\pi\approx64\times3.14 = 200.96\) square inches.

Step3: Analyze the symmetry and find the area of the shaded region

By symmetry, the area of the shaded region \(A\) is equal to half of the area of the square.
\(A=\frac{1}{2}A_{square}\)

Answer:

A. 64