Question

the circle has center o. its radius is 3 ft, and the central angle a measures 90°. what is the area of the shaded region? give the exact answer in terms of π, and be sure to include the correct unit in your answer.

Explanation:

Step1: Recall the formula for the area of a sector

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 measure in degrees and $r$ is the radius of the circle.

Step2: Identify the values of $\theta$ and $r$

We are given that $r = 3$ ft and $\theta=90^{\circ}$.

Step3: Substitute the values into the formula

Substitute $\theta = 90^{\circ}$ and $r = 3$ ft into the formula $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$. We get $A=\frac{90^{\circ}}{360^{\circ}}\times\pi\times(3)^{2}$.

Step4: Simplify the expression

First, $\frac{90^{\circ}}{360^{\circ}}=\frac{1}{4}$, and $(3)^{2}=9$. Then $A=\frac{1}{4}\times\pi\times9=\frac{9\pi}{4}$ square feet.

Answer:

$\frac{9\pi}{4}\text{ ft}^2$