Answer:
The area of the glass in region BCIH is 11 to the nearest feet²
Step-by-step explanation:
* Lets explain the figure
- The window is a semicircle with center G and radius 6 feet
- There is a small semicircle with center G and radius GF
∵ GE is 6 feet and EF is 3 feet
∵ GE = GF + FE
∴ 6 = GF + 3 ⇒ subtract 3 from both sides
∴ 3 = GF
∴ The radius of the small semicircle is 3 feet
∵ m∠BGC = 45°
- The area of sector BGC is part of the area of the semicircle
∵ The area of semi-circle is 1/2 π r²
∵ The measure of the central angle of the semicircle is 180°
∵ The measure of the central angle of the sector BGC is 45°
∴ The sector = 45°/180° = 1/4 of the semi-circle
∴ The area of the sector is 1/4 the area of the semicircle
∵ The area of the semicircle = 1/2 π r²
∵ r = 6 feet
∴ The area of the semicircle = 1/2 π (6)² = 1/2 π (36) = 18 π feet²
∴ Area of the sector = 1/4 (18 π) = 4.5 π feet²
- The small sector HGI has the same central angle of the sector BGC
∴ The area of the sector HGI is 1/4 The area of the small semicircle
∵ The area of the small semicircle = 1/2 π r²
∵ r = 3 feet
∴ The area of the small semicircle = 1/2 π (3)² = 1/2 π (9) = 4.5 π feet²
∴ Area of the sector HGI= 1/4 (4.5 π) = 1.125 π feet²
- The area of the glass in region BCIH is the difference between the
area of sector BGC and the area of the sector HGI
∴ The area of the glass in region BCIH = 4.5 π - 1.125 π ≅ 11 feet²
