Question

1. (8 points) reminders: ( a=\frac{1}{2} r^{2} \theta, s=r \theta ). find the following:

a. if the central angle is ( \theta=75^{circ} ) and the radius of the circle is ( 20 mathrm{~cm} ), what is the arc length ( s ) ? leave your
answer in terms of ( pi ).
b. if the central angle is ( \theta=45^{circ} ) and the area of the sector is ( 50 pi mathrm{ft}^{2} ), what is the radius of the circle?

Explanation:

Step1: Convert angle to radians (for part A)

To use the formula \(s = r\theta\) (where \(\theta\) is in radians), convert \(75^{\circ}\) to radians.
\(\theta = 75\times\frac{\pi}{180}=\frac{5\pi}{12}\) radians.

Step2: Calculate arc length (for part A)

Given \(r = 20\) cm and \(\theta=\frac{5\pi}{12}\), use \(s=r\theta\).
\(s = 20\times\frac{5\pi}{12}=\frac{100\pi}{12}=\frac{25\pi}{3}\) cm.

Step3: Use area formula (for part B)

Given \(A = 50\pi\) ft² and \(\theta = 45^{\circ}=45\times\frac{\pi}{180}=\frac{\pi}{4}\) radians.
From \(A=\frac{1}{2}r^{2}\theta\), substitute values: \(50\pi=\frac{1}{2}r^{2}\times\frac{\pi}{4}\).

Step4: Solve for \(r\) (for part B)

First, simplify the equation \(50\pi=\frac{\pi r^{2}}{8}\).
Multiply both sides by \(8\): \(400\pi=\pi r^{2}\).
Divide both sides by \(\pi\): \(r^{2} = 400\).
Take square root: \(r = 20\) ft.

Answer:

A. \(\frac{25\pi}{3}\) cm
B. \(20\) ft