Question

which of the following formulas is used to calculate the length of an arc in radians?
a. l = 2πθ
b. l = θr²
c. l = 1/2θr²
d. l = θr

if a sector has an arc length of 8π meters and a radius of 8 meters, what is the central angle in radians?
a. 2 radians
b. π radians
c. 2π radians
d. π/2 radians

how do you convert an angle from degrees to radians?
a. multiply by π and divide by 180
b. divide by 360 and multiply by π
c. divide by π and multiply by 180
d. multiply by 360 and divide by π

Explanation:

Step1: Recall arc - length formula

The formula for the length of an arc $L$ in radians is $L = r\theta$, where $r$ is the radius of the circle and $\theta$ is the central - angle in radians. So the answer to the first question is d.

Step2: Use arc - length formula to find central angle

Given $L = 8\pi$ meters and $r = 8$ meters, and using the formula $L=r\theta$. We can solve for $\theta$ by rearranging the formula to $\theta=\frac{L}{r}$. Substituting the given values, we get $\theta=\frac{8\pi}{8}=\pi$ radians. So the answer to the second question is b.

Step3: Recall degree - to - radian conversion

To convert an angle from degrees to radians, we use the conversion factor that $180^{\circ}=\pi$ radians. So if we have an angle $\alpha$ in degrees and want to convert it to radians $\beta$, we use the formula $\beta=\frac{\alpha\times\pi}{180}$. That is, we multiply the angle in degrees by $\pi$ and divide by 180. So the answer to the third question is a.

Answer:

1. d. $L = \theta r$
2. b. $\pi$ radians
3. a. Multiply by $\pi$ and divide by 180