A circle has a chord that goes from #( 3 pi)/8 # to #(4 pi) / 3 # 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 #( 3 pi)/8  # to #(4  pi) /  3  # radians on the circle.  If the area of the circle is #48   pi  #, what is the length of the chord?

Aaliyah Anderson · Accepted Answer

#13.83#

To start, we know that the area of a circle is equal it the radius square times pi.

#A=r^2xxpi#

We also know the area of the circle is #48pi#, so using this we know that

#48pi=r^2xxpi#

We can divide through by pi.

#48=r^2#

And square root.

#4sqrt3=r#

We have calculated the radius of the circle.

Now to find the angle across our chord we subtract the two angles we have been given.

#theta=(4pi)/3-(3pi)/8=(23pi)/24#

Theodore Abad · Answer

undefined To find the length of 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 and $ \theta $ is the angle subtended by the chord at the center of the circle.

Given that the area of the circle is $ 48\pi $, you can use the formula for the area of a circle:

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

to find the radius $ r $.

Once you have the radius $ r $, you can calculate the length of the chord using the formula for chord length mentioned above, with the angle $ \theta $ given in radians as $ \left(\frac{4\pi}{3} - \frac{3\pi}{8}\right) $.