A circle has a chord that goes from #pi/4 # to #pi/8 # radians on the circle. If the area of the circle is #48 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #pi/4  # to #pi/8  # radians on the circle.  If the area of the circle is #48   pi  #, what is the length of the chord?

Sadie Allen · Accepted Answer

#color(purple)("Length of Chord " d = 2r sin theta ~~ 5.3 " units"##

Given #" Area = 48 pi, theta = pi / 4 - pi/ 8 = pi / 8#

#"Chord length " d = 2r * sin theta#

Given #"Area of circle " = pi r^2 = 48pi#

#r = sqrt46 = 4 sqrt3#

#"Length of Chord " d = 2r * sin theta = 2 * 4 sqrt 3 * sin (pi / 8)#

#color(purple)(d = 8 sqrt 3 sin (pi/8) ~~ 5.3 " units"# #

Aaliyah Anderson · Answer

undefined To find the length of the chord in a circle when the area of the circle and the angle subtended by the chord at the center of the circle are given, you can use the following steps:

1. First, determine the radius of the circle using the formula for the area of a circle:
   $$A = \pi r^2$$
   $$48\pi = \pi r^2$$
   $$r^2 = 48$$
   $$r = \sqrt{48} = 4\sqrt{3}$$

2. Next, use the radius and the angles given to find the length of the chord using the formula for the chord length in terms of the radius and the central angle:
   $$l = 2r\sin\left(\frac{\theta}{2}\right)$$
   $$l = 2(4\sqrt{3})\sin\left(\frac{\pi/4 - \pi/8}{2}\right)$$
   $$l = 8\sqrt{3}\sin\left(\frac{\pi}{8}\right)$$

3. Since $\sin(\pi/8)$ is not a common angle, you can use trigonometric identities to express it in terms of more common angles. However, expressing it in a simpler form may not yield a simpler expression for the chord length. Therefore, the final answer for the length of the chord is:
   $$l = 8\sqrt{3}\sin\left(\frac{\pi}{8}\right)$$