A circle has a chord that goes from #pi/3 # to #pi/8 # radians on the circle. If the area of the circle is #25 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #pi/3  # to #pi/8  # radians on the circle.  If the area of the circle is #25   pi  #, what is the length of the chord?

Savannah Aguilar · Accepted Answer

#5 sqrt 2 sqrt {1 - cos frac{5 pi}{24}}# Put #x# and #y# axes adequately so that #x^2 + y^2 = R^2# Area = #pi R^2 = 25 pi Rightarrow R = 5# The chord is #AB#, such that #A = 5 (cos frac{pi}{8}, sin frac{pi}{8})# #B = 5 (cos frac{pi}{3}, sin frac{pi}{3})# #|AB|^2 = (x_A - x_B)^2 + (y_A - y_B)^2# # = 25 (cos a - cos b)^2 + 25 (sin a - sin b)^2# # = 25 (cos^2 a + cos^2 b - 2 cos a cos b + sin^2 a + sin^2 b - 2 sin a sin b)# # = 25 [1 + 1 - 2 cos(a - b) ]# # = 50 [1 - cos(frac{pi}{3} - frac{pi}{8}) ]# #|AB| = 5 sqrt 2 sqrt {1 - cos (pi (8 - 3)/24)}#

Ella Armstrong · Answer

undefined To find the length of the chord, we can use the formula:

$$ \text{Chord Length} = 2r \sin\left(\frac{\theta}{2}\right) $$

Where:
- $ r $ is the radius of the circle.
- $ \theta $ is the angle subtended by the chord at the center of the circle.

Given:
- $ \theta = \frac{\pi}{3} - \frac{\pi}{8} $
- The area of the circle is $ 25\pi $

We know that the area of a circle is given by the formula $ A = \pi r^2 $.
So, if the area of the circle is $ 25\pi $, then:

$$ \pi r^2 = 25\pi $$
$$ r^2 = 25 $$
$$ r = 5 $$

Now, we can substitute the values of $ r $ and $ \theta $ into the formula for the chord length:

$$ \text{Chord Length} = 2(5) \sin\left(\frac{\frac{\pi}{3} - \frac{\pi}{8}}{2}\right) $$

$$ \text{Chord Length} = 10 \sin\left(\frac{\frac{\pi}{3} - \frac{\pi}{8}}{2}\right) $$

$$ \text{Chord Length} = 10 \sin\left(\frac{\frac{5\pi}{24}}{2}\right) $$

$$ \text{Chord Length} = 10 \sin\left(\frac{5\pi}{48}\right) $$

Now, using a calculator to find the value of $ \sin\left(\frac{5\pi}{48}\right) $, and then multiplying it by 10, we can determine the length of the chord.