Question

find the perimeter and total area of the composite shape shown below. all measurements are given in inches. use π = 3.14 in any formulas used. options: perimeter = 52.56 inches, area = 137.12 square inches; perimeter = 46.28 inches, area = 137.12 square inches; perimeter = 65.12 inches, area = 162.24 square inches; perimeter = 52.56 inches, area = 162.24 square inches

Explanation:

Step1: Analyze the composite shape

The composite shape consists of a trapezoid and a semicircle. First, we calculate the perimeter and area separately for each part and then combine them.

Perimeter Calculation:

• The trapezoid has two slant sides of length 10 inches each, a bottom base of 20 inches, and the top base is the diameter of the semicircle (since the radius of the semicircle is 4 inches, the diameter is \( 2\times4 = 8 \) inches). But for the perimeter, we do not include the top base of the trapezoid (where the semicircle is attached) because it is internal.
• The semicircle's arc length is given by \( \frac{1}{2}\times2\pi r=\pi r \). With \( r = 4 \) inches and \( \pi=3.14 \), the arc length is \( 3.14\times4 = 12.56 \) inches.
• Now, sum the lengths of the outer sides: two slant sides (10 each), the bottom base (20), and the semicircle arc (12.56).

\[
\text{Perimeter}=10 + 20+10 + 12.56=52.56 \text{ inches}
\]

Area Calculation:

• Area of Trapezoid: The formula for the area of a trapezoid is \( \frac{(a + b)}{2}\times h \), where \( a \) and \( b \) are the two parallel sides, and \( h \) is the height. Here, \( a = 8 \) inches (top base, diameter of semicircle), \( b = 20 \) inches (bottom base), and \( h = 8 \) inches (height of trapezoid).

\[
\text{Area of Trapezoid}=\frac{(8 + 20)}{2}\times8=\frac{28}{2}\times8 = 14\times8=112 \text{ square inches}
\]

• Area of Semicircle: The formula for the area of a full circle is \( \pi r^{2} \), so the area of a semicircle is \( \frac{1}{2}\pi r^{2} \). With \( r = 4 \) inches and \( \pi = 3.14 \):

\[
\text{Area of Semicircle}=\frac{1}{2}\times3.14\times4^{2}=\frac{1}{2}\times3.14\times16 = 3.14\times8 = 25.12 \text{ square inches}
\]

• Total Area: Sum the area of the trapezoid and the semicircle.

\[
\text{Total Area}=112+25.12 = 137.12 \text{ square inches}
\]

Answer:

Perimeter = 52.56 inches, Area = 137.12 square inches (corresponding to the first option: Perimeter = 52.56 inches, Area = 137.12 square inches)