A circle has a chord that goes from #pi/3 # to #pi/8 # radians on the circle. If the area of the circle is #48 pi #, what is the length of the chord?

1 Answer
Douglas K.
Nov 5, 2016

The chord, #c ~~ 4.45#

Explanation:

Given the Area, A, of the circle is #48pi#, allows us to solve for the radius, using the equation:

#A = pir^2#

#48pi = pir^2#

#r^2 = 48#

Two radii and the chord form a triangle with the angle, #theta = pi/8 - pi/3#, between the two radii. This allows us to use the Law of Cosines to find the length of the chord, c:

#c^2 = r^2 + r^2 - 2(r)(r)cos(theta)#

#c^2 = r^2 + r^2 - 2(r^2)cos(theta)#

#c^2 = 2r^2(1 - cos(theta))#

#c^2 = 96(1 - cos(pi/8 - pi/3))#

#c = sqrt(96(1 - cos(pi/8 - pi/3)))#

#c ~~ 4.45#

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