Question

1. a sector of a circle has central angle 36° and arc length 7.8 cm. find its area to the nearest square centimeter.

Explanation:

Step1: Convert angle to radians

First, convert the central - angle from degrees to radians. We know that $36^{\circ}=36\times\frac{\pi}{180}=\frac{\pi}{5}$ radians.

Step2: Find the radius of the circle

The formula for arc - length $l = r\theta$ (where $l$ is the arc - length, $r$ is the radius, and $\theta$ is the central angle in radians). Given $l = 7.8$ cm and $\theta=\frac{\pi}{5}$, we can solve for $r$. Rearranging the formula gives $r=\frac{l}{\theta}$. Substituting the values, we have $r=\frac{7.8}{\frac{\pi}{5}}=\frac{7.8\times5}{\pi}=\frac{39}{\pi}$ cm.

Step3: Calculate the area of the sector

The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$. Substitute $r = \frac{39}{\pi}$ and $\theta=\frac{\pi}{5}$ into the formula:
\[

$$\begin{align*} A&=\frac{1}{2}\times(\frac{39}{\pi})^{2}\times\frac{\pi}{5}\\ &=\frac{1}{2}\times\frac{39^{2}}{\pi^{2}}\times\frac{\pi}{5}\\ &=\frac{1}{2}\times\frac{39^{2}}{5\pi}\\ &=\frac{1}{2}\times\frac{1521}{5\pi}\\ &=\frac{1521}{10\pi}\\ &\approx\frac{1521}{10\times3.14}\\ &=\frac{1521}{31.4}\\ &\approx48.44 \end{align*}$$

\]

Answer:

$48$ $cm^{2}$