Question

the window shown is the shape of a semicircle with a radius of 6 feet. the distance from f to e is 3 feet and the measure of arc bc = 45°. find the area of the glass in region bcih, rounded to the nearest square foot. ft²

Explanation:

Step1: Find the area formula for a sector of a circle

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 and $r$ is the radius of the circle.

Step2: Determine the outer - radius and inner - radius

The outer - radius $R = 6$ feet and the inner - radius $r=6 - 3=3$ feet. The central angle $\theta = 45^{\circ}$.

Step3: Calculate the area of the outer - sector

For the outer - sector with radius $R = 6$ feet and $\theta = 45^{\circ}$, using the sector area formula $A_{1}=\frac{45^{\circ}}{360^{\circ}}\times\pi R^{2}=\frac{45}{360}\times\pi\times6^{2}=\frac{1}{8}\times\pi\times36=\frac{9\pi}{2}$ square feet.

Step4: Calculate the area of the inner - sector

For the inner - sector with radius $r = 3$ feet and $\theta = 45^{\circ}$, using the sector area formula $A_{2}=\frac{45^{\circ}}{360^{\circ}}\times\pi r^{2}=\frac{45}{360}\times\pi\times3^{2}=\frac{1}{8}\times\pi\times9=\frac{9\pi}{8}$ square feet.

Step5: Calculate the area of the region $BCIH$

The area of the region $BCIH$ is $A = A_{1}-A_{2}$.
\[

$$\begin{align*} A&=\frac{9\pi}{2}-\frac{9\pi}{8}\\ &=\frac{36\pi - 9\pi}{8}\\ &=\frac{27\pi}{8}\\ &\approx\frac{27\times3.14}{8}\\ &=\frac{84.78}{8}\\ & = 10.5975\approx11 \end{align*}$$

\]

Answer:

$11$