Question

a) what is the exact perimeter of the figure?
b) what is the approximate perimeter of the figure, rounded to two decimal places?

Explanation:

Part a)

Step1: Identify the components of the figure. The figure has two sides of 8 in, one side of 14 in, and a semicircular arc. The diameter of the semicircle is 8 in, so the radius \( r=\frac{8}{2} = 4 \) in.

Step2: Calculate the length of the semicircular arc. The formula for the circumference of a full circle is \( C = 2\pi r \), so for a semicircle, it's \( \frac{1}{2} \times 2\pi r=\pi r \). Substituting \( r = 4 \), we get \( \pi\times4 = 4\pi \) in.

Step3: Sum up all the sides. The two 8 - in sides, the 14 - in side, and the semicircular arc. So the perimeter \( P=8 + 8+14 + 4\pi \). Simplifying the linear terms: \( 8 + 8+14=30 \). So the exact perimeter is \( 30 + 4\pi \) inches.

Step1: We know from part a) that the perimeter is \( P = 30+4\pi \). We use the approximation \( \pi\approx3.14159 \).

Step2: Calculate \( 4\pi \approx4\times3.14159 = 12.56636 \).

Step3: Add this to 30: \( 30+12.56636 = 42.56636 \).

Step4: Round to two decimal places. \( 42.56636\approx42.57 \).

Answer:

The exact perimeter of the figure is \( 30 + 4\pi \) inches.

Part b)