Question

1. a sector of a circle has area 9 cm² and a central angle 0.5 radians. find its radius and arc length.

Explanation:

Step1: Recall the area formula for a sector

The area formula of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$, where $A$ is the area of the sector, $r$ is the radius of the circle, and $\theta$ is the central - angle in radians. Given $A = 9\ cm^{2}$ and $\theta=0.5$ radians. We substitute these values into the formula:
$9=\frac{1}{2}r^{2}(0.5)$

Step2: Solve for the radius $r$

First, rewrite the equation from Step 1:
$9=\frac{1}{2}r^{2}(0.5)=\ 0.25r^{2}$.
Then, solve for $r^{2}$: $r^{2}=\frac{9}{0.25}=36$.
Taking the square root of both sides (since $r>0$ as it represents a length), we get $r = 6\ cm$.

Step3: Recall the arc - length formula

The arc - length formula of a sector of a circle is $s=r\theta$, where $s$ is the arc - length, $r$ is the radius, and $\theta$ is the central angle in radians.
We know $r = 6\ cm$ and $\theta=0.5$ radians. Substitute these values into the formula:
$s=(6)(0.5)=3\ cm$.

Answer:

Radius: $6\ cm$, Arc length: $3\ cm$