Question

the length of an arc of a circle is 7.34 units, and the measure of the corresponding central - angle is 81°. what is the approximate length of the radius of the circle?
a. 10.36 units
b. 5.19 units
c. 3.59 units
d. 11.03 units

Explanation:

Step1: Convert angle to radians

First, convert the central - angle from degrees to radians. The formula to convert degrees to radians is $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg} = 81^{\circ}$, then $\theta_{rad}=81\times\frac{\pi}{180}=\frac{9\pi}{20}$ radians.

Step2: Use arc - length formula

The arc - length formula is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius, and $\theta$ is the central angle in radians. We know $s = 7.34$ and $\theta=\frac{9\pi}{20}$. Rearranging the formula for $r$, we get $r=\frac{s}{\theta}$.

Step3: Calculate the radius

Substitute $s = 7.34$ and $\theta=\frac{9\pi}{20}$ into the formula for $r$:
\[

$$\begin{align*} r&=\frac{7.34}{\frac{9\pi}{20}}\\ &=7.34\times\frac{20}{9\pi}\\ &=\frac{146.8}{9\pi}\\ &\approx\frac{146.8}{9\times3.14}\\ &=\frac{146.8}{28.26}\\ &\approx5.19 \end{align*}$$

\]

Answer:

B. 5.19 units