Question

what is the area of the sector bound by the center of the circle and \\(\overarc{cd}\\) in the circle below? (image of a circle with center a, radius r = 15 ft, angle between ac and ad is 45 degrees) \\(\circ\\) a 9.42 ft squared \\(\circ\\) b 19.54 ft squared \\(\circ\\) c 34.89 ft squared \\(\circ\\) d 88.31 ft squared \\(\circ\\) e i have not learned this yet.

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is \( A=\frac{\theta}{360^{\circ}}\times\pi r^{2} \).

Step2: Identify the given values

We are given that \( r = 15\space\text{ft} \) and \( \theta=45^{\circ} \).

Step3: Substitute the values into the formula

First, calculate \( r^{2} \): \( r^{2}=15^{2} = 225 \).
Then, substitute into the sector area formula: \( A=\frac{45^{\circ}}{360^{\circ}}\times\pi\times225 \).
Simplify \( \frac{45}{360}=\frac{1}{8} \), so \( A = \frac{1}{8}\times\pi\times225 \).
Calculate \( \frac{225}{8}\times\pi \approx\frac{225}{8}\times 3.1416 \).
\( \frac{225}{8}=28.125 \), then \( 28.125\times3.1416\approx88.31 \space\text{ft}^2 \).

Answer:

d. 88.31 ft squared