A circle has a chord that goes from #( 5 pi)/3 # to #(17 pi) / 12 # radians on the circle. If the area of the circle is #18 pi #, what is the length of the chord?

Question

A circle has a chord that goes from #( 5 pi)/3  # to #(17  pi) /  12  # radians on the circle.  If the area of the circle is #18   pi  #, what is the length of the chord?

Mason Adams · Accepted Answer

The length of the chord is #c ~~ 3.25#

We can compute the radius, given the area of the circle:

#A = pir^2#

#18pi = pir^2#

#18 = r^2#

#sqrt18 = r#

Compute the angle:

#theta = (17pi)/12 - (5pi)/3#

#theta = (17pi)/12 - (20pi)/12#

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

We can compute the length of the chord, c, using the Law of Cosines:

#c^2 = a^2 + b^2 - 2(a)(b)cos(theta)

where # a = b = r# and #theta = -pi/4#:

#c^2 = r^2 + r^2 - 2(r)(r)cos(-pi/4)#

#c^2 = 36(1 - sqrt(2)/2)#

#c ~~ 3.25#

Savannah Aguilar · Answer

undefined The length of the chord can be calculated using 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 $ 18\pi $, we can find the radius $ r $ using the formula for the area of a circle:

$$ A = \pi r^2 $$

Substituting $ A = 18\pi $, we find:

$$ 18\pi = \pi r^2 $$
$$ r^2 = 18 $$
$$ r = \sqrt{18} = 3\sqrt{2} $$

Now, we need to find the angle $ \theta $ subtended by the chord. The difference between the given angles gives us the measure of $ \theta $:

$$ \theta = \frac{17\pi}{12} - \frac{5\pi}{3} $$
$$ \theta = \frac{17\pi}{12} - \frac{20\pi}{12} $$
$$ \theta = \frac{-3\pi}{12} = -\frac{\pi}{4} $$

However, since $ \theta $ should be positive, we take its absolute value:

$$ \theta = \left|\frac{-\pi}{4}\right| = \frac{\pi}{4} $$

Now, using the formula for the length of the chord:

$$ \text{Chord length} = 2 \cdot 3\sqrt{2} \cdot \sin\left(\frac{\pi}{4}\right) $$
$$ \text{Chord length} = 6\sqrt{2} \cdot \frac{\sqrt{2}}{2} $$
$$ \text{Chord length} = 6 $$

So, the length of the chord is $ 6 $ units.