A circle has a chord that goes from #pi/3 # to #pi/4 # radians on the circle. If the area of the circle is #49 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #pi/3  # to #pi/4  # radians on the circle.  If the area of the circle is #49   pi  #, what is the length of the chord?

Wyatt Anderson · Accepted Answer

The length of the chord is #=1.83u#

The angle subtended at the center of the circle is

#hat(AOB)=theta=pi/3-pi/4=pi/12#

The area of the circle is

#A=pir^2=49pi#

Therefore,

The radius of the circle is

#r=sqrt49=7#

The length of the chord is

#AB=2*AC=2*AOsin(theta/2)#

#=2*7*sin(pi/24)#

#=14*0.13#

#=1.83u#

Sophia Alfred · Answer

undefined To find the length of the chord in a circle given the area of the circle and the central angle subtended by the chord, you 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 central angle subtended by the chord.

Given that the area of the circle is $ 49\pi $, we can find the radius $ r $ using the formula for the area of a circle:

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

$$ 49\pi = \pi r^2 $$

$$ r^2 = 49 $$

$$ r = 7 $$

Now, we can use the given central angle to find the length of the chord:

$$ \theta = \frac{\pi}{3} - \frac{\pi}{4} = \frac{\pi}{12} $$

$$ \text{Chord Length} = 2(7) \sin\left(\frac{\pi}{12}\right) $$

$$ \text{Chord Length} \approx 2(7) \times 0.2588 $$

$$ \text{Chord Length} \approx 9.128 $$

So, the length of the chord is approximately $ 9.128 $ units.