Question
FOG is a sector of a circle, centre O, with angle FOG = 1.2 radians. EOH a sector of a circle, centre O, with radius 5 cm.
The shaded region has an area of 71.4c * m ^ 2
Find the perimeter of the shaded region.
The shaded region has an area of 71.4c * m ^ 2
Find the perimeter of the shaded region.
Answers
GPT 3.5
To find the perimeter of the shaded region, we need to first find the radius of the sector FOG.
Since the angle FOG is 1.2 radians and the circumference of a full circle is 2πr, the circumference of the sector FOG can be calculated as:
circumference of sector FOG = (1.2 / 2π) * 2π * r = 1.2r
Since the circumference is equal to the arc length of the sector, we have:
1.2r = 5
r = 5 / 1.2 = 25 / 6 ≈ 4.17 cm
The perimeter of the shaded region consists of the arc length of the sector FOG and the segment EH. The arc length of the sector FOG is 1.2 * 4.17 ≈ 5.00 cm.
To find the length of segment EH, we can use the fact that the area of the sector EOH is 71.4 cm^2. The ratio of the area of the sector EOH to the area of the full circle is equal to the ratio of the angle EOH to 2π radians:
(area of sector EOH) / (π * 5^2) = angle EOH / 2π
71.4 / 25π = angle EOH / 2π
angle EOH = 2π * 71.4 / 25π = 4.54 radians
The length of segment EH can now be calculated as:
length of EH = 5 * 4.54 / 2π = 3.64 cm (approximately)
Therefore, the total perimeter of the shaded region is approximately 5.00 cm (arc length of sector FOG) + 3.64 cm (length of segment EH) = 8.64 cm.
Since the angle FOG is 1.2 radians and the circumference of a full circle is 2πr, the circumference of the sector FOG can be calculated as:
circumference of sector FOG = (1.2 / 2π) * 2π * r = 1.2r
Since the circumference is equal to the arc length of the sector, we have:
1.2r = 5
r = 5 / 1.2 = 25 / 6 ≈ 4.17 cm
The perimeter of the shaded region consists of the arc length of the sector FOG and the segment EH. The arc length of the sector FOG is 1.2 * 4.17 ≈ 5.00 cm.
To find the length of segment EH, we can use the fact that the area of the sector EOH is 71.4 cm^2. The ratio of the area of the sector EOH to the area of the full circle is equal to the ratio of the angle EOH to 2π radians:
(area of sector EOH) / (π * 5^2) = angle EOH / 2π
71.4 / 25π = angle EOH / 2π
angle EOH = 2π * 71.4 / 25π = 4.54 radians
The length of segment EH can now be calculated as:
length of EH = 5 * 4.54 / 2π = 3.64 cm (approximately)
Therefore, the total perimeter of the shaded region is approximately 5.00 cm (arc length of sector FOG) + 3.64 cm (length of segment EH) = 8.64 cm.
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