Question

a rose garden is formed by joining a rectangle and a semicircle, as shown below. the rectangle is 20 ft long and 16 ft wide. find the area of the garden. 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 rectangle

The formula for the area of a rectangle is \( A_{rectangle} = length \times width \). Given length = 20 ft and width = 16 ft, so \( A_{rectangle} = 20 \times 16 = 320 \) square feet.

Step2: Calculate area of semicircle

The diameter of the semicircle is equal to the width of the rectangle, so diameter \( d = 16 \) ft, radius \( r = \frac{d}{2} = \frac{16}{2} = 8 \) ft. The formula for the area of a full circle is \( A_{circle} = \pi r^2 \), so the area of a semicircle is \( A_{semicircle} = \frac{1}{2} \pi r^2 \). Substituting \( \pi = 3.14 \) and \( r = 8 \), we get \( A_{semicircle} = \frac{1}{2} \times 3.14 \times 8^2 = \frac{1}{2} \times 3.14 \times 64 = 100.48 \) square feet.

Step3: Calculate total area of garden

The total area of the garden is the sum of the area of the rectangle and the area of the semicircle. So \( A_{total} = A_{rectangle} + A_{semicircle} = 320 + 100.48 = 420.48 \) square feet.

Answer:

\( 420.48 \) square feet (or \( 420.48\ \text{ft}^2 \))