A circle has a chord that goes from #( pi)/8 # to #(5 pi) / 6 # radians on the circle. If the area of the circle is #225 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #(  pi)/8  # to #(5  pi) /  6  # radians on the circle.  If the area of the circle is #225   pi  #, what is the length of the chord?

Aurora Ayala · Accepted Answer

#=33.38# A chord that goes from #(pi)/8#to #(5pi)/6# so it travels the distance #(5pi)/6-(pi)/8=(17pi)/24#; or #(17pi)/24-:2pi=17/48# of the Circumference of the Circle Area of the Circle#=pir^2=225pi# or #r^2=225# or #r=sqrt225# or #r=15# Circumference of the circle#=2pir=2(pi)(15)=94.24# Therefore Length of the chord#=94.24times17/48=33.38#

Wyatt Anderson · Answer

undefined 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 $ r $ is the radius of the circle and $ \theta $ is the angle subtended by the chord at the center of the circle.

First, find the radius of the circle using the formula for the area of a circle:

$$ \text{Area of Circle} = \pi r^2 $$

Given that the area of the circle is $ 225 \pi $, you can solve for $ r $.

Next, find the length of the chord using the given angles $ \frac{\pi}{8} $ and $ \frac{5\pi}{6} $ to determine the angle $ \theta $. Then, substitute the values into the formula for the length of the chord.