# A circle has a chord that goes from #( 5 pi)/6 # to #(5 pi) / 4 # radians on the circle. If the area of the circle is #21 pi #, what is the length of the chord?

Let c = the length of the chord; its length can found using a variant of the Law of Cosines:

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To find the length of the chord, we first need to find the radius of the circle. We know that the area of a circle is given by the formula (A = \pi r^2), where (A) is the area and (r) is the radius.

Given that the area of the circle is (21\pi), we can set up the equation as follows:

[21\pi = \pi r^2]

Solving for (r), we get:

[r^2 = \frac{21\pi}{\pi}] [r^2 = 21]

Taking the square root of both sides:

[r = \sqrt{21}]

Now that we have the radius, we can use it to find the length of the chord. The length of the chord in a circle is given by the formula (2r\sin(\frac{\theta}{2})), where (r) is the radius and (\theta) is the angle subtended by the chord at the center of the circle.

Given that the angle (\theta) is from (\frac{5\pi}{6}) to (\frac{5\pi}{4}), we can find the difference between these two angles to get the measure of (\theta):

[\theta = \frac{5\pi}{4} - \frac{5\pi}{6}] [\theta = \frac{\pi}{2}]

Now, we can plug in the values into the formula:

[2(\sqrt{21})\sin\left(\frac{\pi}{4}\right)]

Using the fact that (\sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}), we get:

[2(\sqrt{21})\left(\frac{\sqrt{2}}{2}\right)] [= \sqrt{21} \times \sqrt{2}] [= \sqrt{42}]

Therefore, the length of the chord is (\sqrt{42}) 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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