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 = 45°. find the area of the glass in region bcih, rounded to the nearest square foot.

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