Question

1. find the area ( a ) of the sector and the arc length ( s ) of a circle of radius 5 feet formed by the central angle ( 45^circ ). (remember that the formulas you hopefully have memorized only work in

Explanation:

Step1: Convert degrees to radians

First, convert the central angle from degrees to radians since sector area and arc length formulas use radians.
$$45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4} \text{ radians}$$

Step2: Calculate sector area $A$

Use the sector area formula $A = \frac{1}{2}r^2\theta$, where $r=5$ ft and $\theta=\frac{\pi}{4}$.
$$A = \frac{1}{2} \times 5^2 \times \frac{\pi}{4} = \frac{1}{2} \times 25 \times \frac{\pi}{4} = \frac{25\pi}{8}$$

Step3: Calculate arc length $S$

Use the arc length formula $S = r\theta$, where $r=5$ ft and $\theta=\frac{\pi}{4}$.
$$S = 5 \times \frac{\pi}{4} = \frac{5\pi}{4}$$

Answer:

Sector area: $\frac{25\pi}{8}$ square feet, Arc length: $\frac{5\pi}{4}$ feet