find the sector area of the entire shaded region. use $pi$ in your calculations and round to the nearest…

Explanation:

Step1: Find the central angle of the shaded region

The central angles of the shaded - regions are $47^{\circ}$, $90^{\circ}$ and $80^{\circ}$. The sum of the central angles of the shaded region $\theta=47 + 90+80=217^{\circ}$.

Step2: Recall the formula for the area of a sector

The formula for the area of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees) is $A=\frac{\theta}{360}\times\pi r^{2}$. Here, $r = 23$ inches.

Step3: Substitute the values into the formula

Substitute $\theta = 217$ and $r = 23$ into the formula: [ \begin{align*} A&=\frac{217}{360}\times\pi\times(23)^{2}\ &=\frac{217}{360}\times\pi\times529\ &=\frac{217\times529\pi}{360}\ &=\frac{114893\pi}{360} \end{align*} ]

Step4: Calculate the value and round

[ \begin{align*} A&=\frac{114893\pi}{360}\approx\frac{114893\times3.14159}{360}\ &=\frac{360027.13487}{360}\approx1000.08 \end{align*} ]

Answer:

$1000.08$ square inches