Question

circle b is shown in the diagram. the shaded sector has an area of 3π square meters. what is the radius of circle b? meters

Explanation:

Step1: Recall sector - area formula

The formula for the area of a sector of a circle is $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $\theta$ is the central - angle of the sector in degrees, $r$ is the radius of the circle, and $A$ is the area of the sector.

Step2: Substitute the given values into the formula

We know that $A = 3\pi$ and $\theta=120^{\circ}$. Substituting these values into the formula $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, we get $3\pi=\frac{120^{\circ}}{360^{\circ}}\times\pi r^{2}$.

Step3: Simplify the equation

First, simplify $\frac{120^{\circ}}{360^{\circ}}=\frac{1}{3}$. The equation becomes $3\pi=\frac{1}{3}\times\pi r^{2}$.

Step4: Solve for $r^{2}$

Multiply both sides of the equation by 3 to get rid of the fraction: $3\pi\times3=\pi r^{2}$, which simplifies to $9\pi=\pi r^{2}$. Then divide both sides by $\pi$: $\frac{9\pi}{\pi}=\frac{\pi r^{2}}{\pi}$, and we obtain $r^{2}=9$.

Step5: Solve for $r$

Take the square - root of both sides. Since $r$ represents the radius of a circle (a non - negative quantity), $r = 3$.

Answer:

3