Question

the following illustration depicts a circle positioned within a square with side lengths of 4 inches. which of the following expressions best represents the area of the shaded regions in the figure above?

Explanation:

Step1: Calculate area of square

The side length of the square is 4 inches. The formula for the area of a square is \( A_{square} = s^2 \), where \( s \) is the side length. So, \( A_{square} = 4^2 = 16 \) square inches.

Step2: Calculate area of circle

The diameter of the circle is equal to the side length of the square, so the diameter \( d = 4 \) inches, and the radius \( r = \frac{d}{2} = \frac{4}{2} = 2 \) inches. The formula for the area of a circle is \( A_{circle} = \pi r^2 \). Substituting \( r = 2 \), we get \( A_{circle} = \pi (2)^2 = 4\pi \) square inches.

Step3: Calculate area of shaded region

The shaded region's area is the area of the square minus the area of the circle. So, \( A_{shaded} = A_{square} - A_{circle} = 16 - 4\pi \) square inches.

Answer:

The area of the shaded region is \( 16 - 4\pi \) square inches (or approximately \( 16 - 12.57 = 3.43 \) square inches if a numerical approximation for \( \pi \) is used).