A circle has a chord that goes from #( pi)/4 # to #(13 pi) / 8 # radians on the circle. If the area of the circle is #27 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #(  pi)/4  # to #(13  pi) /  8  # radians on the circle.  If the area of the circle is #27   pi  #, what is the length of the chord?

Aaliyah Anderson · Accepted Answer

Length of the chord#=22.45# A chord that goes from #(pi)4#to #(13pi)/8# so it travels the distance #(13pi)/8-(pi)/4=(11pi)/8#; or #(11pi)/8-:2pi=11/16# of the Circumference of the Circle Area of the Circle#=pir^2=27pi# or #r^2=27# or #r=sqrt27# or #r=5.2# Circumference of the circle#=2pir=2(pi)(5.2)=32.65# Therefore Length of the chord#=32.65times11/16=22.45#

Evelyn Arboleda · Answer

undefined To find the length of the chord, you can use the formula for the area of a circle and the properties of chords in circles.

1. The area of a circle is given by $ A = \pi r^2 $, where $ A $ is the area and $ r $ is the radius of the circle.

2. Given that the area of the circle is $ 27 \pi $, you can solve for the radius:
   $ 27 \pi = \pi r^2 $
   $ r^2 = 27 $
   $ r = \sqrt{27} = 3\sqrt{3} $

3. Now, to find the length of the chord, you can use the formula for the length of a chord in a circle:
   $ \text{Chord Length} = 2r \sin\left(\frac{\theta}{2}\right) $,
   where $ \theta $ is the angle subtended by the chord at the center of the circle.

4. The angle subtended by the chord is $ \frac{13\pi}{8} - \frac{\pi}{4} = \frac{13\pi - 2\pi}{8} = \frac{11\pi}{8} $.

5. Substitute the values into the formula:
   $ \text{Chord Length} = 2 \times 3\sqrt{3} \sin\left(\frac{11\pi}{16}\right) $

6. Calculate the length:
   $ \text{Chord Length} \approx 2 \times 3\sqrt{3} \times 0.83147 $ (approximately)
   $ \text{Chord Length} \approx 4.989 \times \sqrt{3} $

So, the length of the chord is approximately $ 4.989 \times \sqrt{3} $ units.