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A sector of a circle has an area of 32πcm^2 and contains a central angle of 210 degrees.calculate the radius of the circle and length of the cord that subtends the angle given above
Bot GPT 3.5
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The area of a sector of a circle is given by the formula:
Area of sector = (θ/360) * π * r^2
Where θ is the central angle in degrees and r is the radius of the circle.
Given that the area of the sector is 32πcm^2 and the central angle is 210 degrees, we can plug these values into the formula and solve for the radius:
32π = (210/360) * π * r^2
32 = (7/12) * r^2
r^2 = 32 / (7/12)
r^2 = 32 * (12/7)
r^2 = 384/7
r ≈ 7.02 cm
Therefore, the radius of the circle is approximately 7.02 cm.
To calculate the length of the chord that subtends the central angle of 210 degrees, we can use the formula:
Chord length = 2 * r * sin(θ/2)
Where r is the radius of the circle and θ is the central angle in radians.
Converting 210 degrees to radians:
210 degrees * (π/180) = 7π/6 radians
Now plug in the values to get:
Chord length = 2 * 7.02 * sin(7π/6)
Chord length = 14.04 * sin(7π/6)
Chord length = 14.04 * (-1/2)
Chord length = -7.02
Therefore, the length of the chord that subtends the central angle of 210 degrees is 7.02 cm.