Question

finding composite area 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 \\(\overarc{bc} = 45^\circ\\). find the area of the glass in region bcih, rounded to the nearest square foot. \\(\square\\) \\(\text{ft}^2\\)

Explanation:

Step1: Determine the radii of the two sectors

The radius of the larger sector (GB) is 6 feet. The radius of the smaller sector (GI) is \( 6 - 3 = 3 \) feet (since the distance from F to E is 3 feet, and GF is equal to GI, GE is equal to GB).

Step2: Recall the formula for the area of a sector

The area of a sector of a circle with radius \( r \) and central angle \( \theta \) (in degrees) is given by \( A=\frac{\theta}{360}\times\pi r^{2} \). For a semicircle, the total angle is \( 180^{\circ} \), but here the central angle for sector BCIH is \( 45^{\circ} \).

Step3: Calculate the area of the larger sector (BCG)

Using the formula for the area of a sector with \( r = 6 \) and \( \theta=45^{\circ} \):
\( A_{1}=\frac{45}{360}\times\pi\times(6)^{2}=\frac{1}{8}\times\pi\times36=\frac{36\pi}{8}=\frac{9\pi}{2} \)

Step4: Calculate the area of the smaller sector (HCI)

Using the formula for the area of a sector with \( r = 3 \) and \( \theta = 45^{\circ} \):
\( A_{2}=\frac{45}{360}\times\pi\times(3)^{2}=\frac{1}{8}\times\pi\times9=\frac{9\pi}{8} \)

Step5: Calculate the area of region BCIH

The area of region BCIH is the area of the larger sector minus the area of the smaller sector:
\( A = A_{1}-A_{2}=\frac{9\pi}{2}-\frac{9\pi}{8}=\frac{36\pi - 9\pi}{8}=\frac{27\pi}{8} \)

Now, we calculate the numerical value:
\( \frac{27\pi}{8}\approx\frac{27\times3.1416}{8}=\frac{84.8232}{8} = 10.6029\approx11 \)

Answer:

11