Question

a triangle is placed in a semicircle with a radius of 8 ft, as shown below. find the area of the shaded region. use the value 3.14 for π, and do not round your answer. be sure to include the correct unit in your answer.

Explanation:

Step1: Calculate area of semicircle

The formula for the area of a full circle is \( A = \pi r^2 \), so the area of a semicircle is \( \frac{1}{2}\pi r^2 \). Given \( r = 8 \) ft and \( \pi = 3.14 \), we substitute these values:
\( \text{Area of semicircle} = \frac{1}{2} \times 3.14 \times 8^2 \)
\( = \frac{1}{2} \times 3.14 \times 64 \)
\( = 3.14 \times 32 \)
\( = 100.48 \) square feet.

Step2: Calculate area of triangle

The triangle is an isosceles triangle with height \( h = 8 \) ft (the radius) and base \( b = 2r = 16 \) ft (the diameter of the semicircle). The formula for the area of a triangle is \( A = \frac{1}{2}bh \). Substituting the values:
\( \text{Area of triangle} = \frac{1}{2} \times 16 \times 8 \)
\( = 8 \times 8 \)
\( = 64 \) square feet.

Step3: Calculate area of shaded region

The shaded region's area is the area of the semicircle minus the area of the triangle:
\( \text{Area of shaded region} = 100.48 - 64 \)
\( = 36.48 \) square feet.

Answer:

\( 36.48 \) square feet (or \( 36.48\ \text{ft}^2 \))