# 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?

The length of the chord is

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

Compute the angle:

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)

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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.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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