SOLUTION: A central angle 𝜃 in a circular garden of radius 20 meters is subtended by an arc of length 15𝜋 meters.
a. What is the measure of 𝜃 in degrees?
b. What is the area of
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Question 1187599: A central angle 𝜃 in a circular garden of radius 20 meters is subtended by an arc of length 15𝜋 meters.
a. What is the measure of 𝜃 in degrees?
b. What is the area of the sector being described?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the radius of the garden is 20 meters.
the circumference of the garden is 2 * pi * r = 40 * pi.
the area of the garden is pi * r^2 = pi * 20^2 = 400 * pi.
the arc is 15 * pi.
the central angle of the arc is 15 * pi / (40 * pi) * 360 = 135 degrees.
the area of the sector is equal to 135/360 * 400 * pi = 150 * pi.
the circumference and the area of the sector of the circle are proportion to the central angle of the sector divided by 360 * the circumference of the circle and also times the area of the circle.
when the central angle of the sector is 135 degrees, then the fraction is 135/360.
the circumference of the circle is 2 * pi * 20 = 40 * pi.
the area of the circle is pi * 20^2 = 400 * pi.
the arc of the sector is 135 / 360 * 40 * pi = 15 * pi.
this is not surprising since we used the circumference of the circle and the length of the arc to find the central angle.
the area of the sector is 135 / 360 * 400 * pi = 150 * pi.
here's a reference on arc length of a sector and area of a sector.
https://byjus.com/maths/sector-of-a-circle/
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