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The perimeter of a sector of a circle of radius 5.7 m is 27.2 m. Find the area of the sector.
- a)90.06 cm2
- b)135.09 cm2
- c)45 cm2
- d)None of these
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
The perimeter of a sector of a circle of radius 5.7 m is 27.2 m. Find ...
Let the angle subtended by the sector at the centre be = q
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Most Upvoted Answer
The perimeter of a sector of a circle of radius 5.7 m is 27.2 m. Find ...
Understanding the Problem
The problem gives us the perimeter of a sector of a circle with a radius of 5.7 m, which is 27.2 m. We need to find the area of the sector.
Components of a Sector's Perimeter
The perimeter of a sector comprises two radii and the arc length. The formula for the perimeter (P) is:
P = 2r + l
Where:
- r = radius
- l = arc length
Given:
- r = 5.7 m
- P = 27.2 m
Finding the Arc Length
We can rearrange the perimeter formula to find the arc length (l):
l = P - 2r
Substituting the values:
l = 27.2 - 2(5.7)
Calculating:
l = 27.2 - 11.4 = 15.8 m
Finding the Angle in Radians
The arc length is related to the radius and the angle (θ) in radians by the formula:
l = r * θ
Rearranging gives us:
θ = l / r
Substituting the values:
θ = 15.8 / 5.7
Calculating the angle:
θ ≈ 2.77 radians
Calculating the Area of the Sector
The area (A) of a sector is given by:
A = (1/2) * r^2 * θ
Substituting the values:
A = (1/2) * (5.7^2) * 2.77
Calculating:
A ≈ 39.2 m²
Since the options given are in cm², we convert:
39.2 m² = 3920 cm²
Conclusion
Since the calculated area (3920 cm²) does not match any given options, the correct answer is indeed option 'D' - None of these.
The problem gives us the perimeter of a sector of a circle with a radius of 5.7 m, which is 27.2 m. We need to find the area of the sector.
Components of a Sector's Perimeter
The perimeter of a sector comprises two radii and the arc length. The formula for the perimeter (P) is:
P = 2r + l
Where:
- r = radius
- l = arc length
Given:
- r = 5.7 m
- P = 27.2 m
Finding the Arc Length
We can rearrange the perimeter formula to find the arc length (l):
l = P - 2r
Substituting the values:
l = 27.2 - 2(5.7)
Calculating:
l = 27.2 - 11.4 = 15.8 m
Finding the Angle in Radians
The arc length is related to the radius and the angle (θ) in radians by the formula:
l = r * θ
Rearranging gives us:
θ = l / r
Substituting the values:
θ = 15.8 / 5.7
Calculating the angle:
θ ≈ 2.77 radians
Calculating the Area of the Sector
The area (A) of a sector is given by:
A = (1/2) * r^2 * θ
Substituting the values:
A = (1/2) * (5.7^2) * 2.77
Calculating:
A ≈ 39.2 m²
Since the options given are in cm², we convert:
39.2 m² = 3920 cm²
Conclusion
Since the calculated area (3920 cm²) does not match any given options, the correct answer is indeed option 'D' - None of these.
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