Question

1. a circle has radius $r = 8$. find the length of the arc determined by a central angle of $\theta=\frac{3pi}{4}$. 10. convert the angle $\theta = 240^{circ}$ to the exact radian value.

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $s$ of a circle with radius $r$ and central - angle $\theta$ (in radians) is $s = r\theta$. Given $r = 8$ and $\theta=\frac{3\pi}{4}$.
$s=8\times\frac{3\pi}{4}$

Step2: Simplify the expression

$8\times\frac{3\pi}{4}=\frac{24\pi}{4}=6\pi$

Step3: Recall degree - to - radian conversion formula

To convert degrees to radians, we use the formula $\text{radians}=\frac{\pi}{180}\times\text{degrees}$. Given $\theta = 240^{\circ}$.
$\theta=\frac{\pi}{180}\times240$

Step4: Simplify the degree - to - radian conversion

$\frac{\pi}{180}\times240=\frac{240\pi}{180}=\frac{4\pi}{3}$

Answer:

1. The length of the arc is $6\pi$.
2. The angle in radians is $\frac{4\pi}{3}$.