Question

How did it come up with 11 as the answer

Answer 1

11 ft²

Explanation:

The area of region BCIH is the area of sector BCG minus the area of sector HIG.

Given that arc BC measures 45°, and a semi-circular arc measures 180°, we can deduce that arc BC represents one-quarter of the semi-circular arc, as 180° ÷ 45° = 4.

Therefore, the area of sector BCG is a quarter of the area of the larger semicircle (with a radius of 6 feet), and the area of sector HIG is a quarter of the area of the smaller semicircle.

Given FE is 3 feet, and the radius of the larger semicircle is 6 feet, the radius of the smaller semicircle is:

`r_(\sf smaller\;semicircle)= 6 - 3 = 3 \sf \; feet`

The area of a semicircle is half the area of a circle.

Therefore, the area of a quarter of a semicircle is 1/8th the area of a circle.

The area of a circle is A = πr². Therefore, the formula for the area of 1/8th of a circle is:

`A=(1)/(8)\pi r^2`

To calculate the area of region BCIH, we can subtract the area of sector HIG (1/8th of a circle with r = 3) from the area of sector BCG (1/8th of a circle with r = 6):

`\begin{aligned}BCIH&=BCG-HIG\\\\&=(1)/(8)\pi (6)^2-(1)/(8)\pi 3^2\\\\&=(1)/(8)\pi (36)-(1)/(8)\pi (9)\\\\&=(36)/(8)\pi -(9)/(8)\pi \\\\&=(27)/(8)\pi \\\\&=10.6028752...\\\\&=11\; \sf ft^2\end{aligned}`

Therefore, the area of the glass in region BCIH is 11 ft², rounded to the nearest square foot.