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 18. (a) 0.8 rad r = (b)

Explanation:

Step1: Recall the 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 the given values into the formula

We are given that $A = 18$ and $\theta=0.8$ rad. Substituting these values into the formula $A=\frac{1}{2}r^{2}\theta$, we get $18=\frac{1}{2}r^{2}(0.8)$.

Step3: Solve for $r^{2}$

First, simplify the right - hand side of the equation: $\frac{1}{2}r^{2}(0.8)=0.4r^{2}$. So, the equation becomes $18 = 0.4r^{2}$. Then, solve for $r^{2}$ by dividing both sides of the equation by $0.4$: $r^{2}=\frac{18}{0.4}=\frac{180}{4} = 45$.

Step4: Solve for $r$

Take the square root of both sides of the equation $r^{2}=45$. Since $r$ represents the radius of a circle (a non - negative quantity), we have $r=\sqrt{45}=3\sqrt{5}\approx6.71$.

Answer:

$r = 3\sqrt{5}$