Question

this exercise involves the formula for the area of a circular sector. find the radius r of each circle if the area of the sector is 24. (a) 0.5 rad

Explanation:

Step1: Recall sector - area formula

The formula for the area of a circular sector 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.

Step2: Substitute given values

We are given that $A = 24$ and $\theta=0.5$ rad. Substituting these values into the formula $A=\frac{1}{2}r^{2}\theta$, we get $24=\frac{1}{2}r^{2}(0.5)$.

Step3: Solve for $r^{2}$

First, simplify the right - hand side of the equation: $\frac{1}{2}r^{2}(0.5)=\frac{0.5}{2}r^{2}=0.25r^{2}$. So, the equation becomes $24 = 0.25r^{2}$. Then, solve for $r^{2}$ by dividing both sides of the equation by $0.25$: $r^{2}=\frac{24}{0.25}=96$.

Step4: Solve for $r$

Take the square root of both sides. Since $r$ represents the radius of a circle and must be non - negative, $r=\sqrt{96}=4\sqrt{6}$.

Answer:

$r = 4\sqrt{6}$