Question

• / 6.66 points consider the following figure. give both exact values and approximations to the nearest hundredth. (a) find the circumference (in ft) of the figure. exact ft approximation ft (b) find the area (in ft²) of the figure. exact ft² approximation ft² (figure: circle with diameter 16 ft)

Explanation:

Part (a): Circumference of the Circle

Step 1: Recall the formula for circumference

The formula for the circumference \( C \) of a circle is \( C = \pi d \), where \( d \) is the diameter. Here, the diameter \( d = 16 \) ft.
\[ C = \pi \times 16 = 16\pi \]

Step 2: Approximate using \( \pi \approx 3.1416 \)

\[ C \approx 16 \times 3.1416 = 50.2656 \approx 50.27 \] (rounded to the nearest hundredth)

Step 1: Find the radius

The radius \( r \) is half of the diameter. So, \( r = \frac{d}{2} = \frac{16}{2} = 8 \) ft.

Step 2: Recall the formula for area

The formula for the area \( A \) of a circle is \( A = \pi r^2 \). Substituting \( r = 8 \):
\[ A = \pi \times 8^2 = 64\pi \]

Step 3: Approximate using \( \pi \approx 3.1416 \)

\[ A \approx 64 \times 3.1416 = 201.0624 \approx 201.06 \] (rounded to the nearest hundredth)

Answer:

(a):
Exact: \( 16\pi \) ft
Approximation: \( 50.27 \) ft

Part (b): Area of the Circle