# A circle has a chord that goes from #( pi)/2 # to #(15 pi) / 8 # radians on the circle. If the area of the circle is #121 pi #, what is the length of the chord?

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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 (121 \pi), we can solve for the radius:

[121 \pi = \pi r^2]

[r^2 = \frac{121 \pi}{\pi}]

[r^2 = 121]

[r = 11]

Now that we have the radius, we can use trigonometry to find the length of the chord. The chord forms an isosceles triangle with the radius, so the angle at the center of the circle is twice the angle at one end of the chord.

The angle at one end of the chord is given by (\frac{15\pi}{8} - \frac{\pi}{2} = \frac{7\pi}{8}).

Thus, the angle at the center of the circle is (2 \times \frac{7\pi}{8} = \frac{7\pi}{4}).

Now, we can use the cosine rule to find the length of the chord:

[ \text{Chord Length} = 2r \sin\left(\frac{\theta}{2}\right) ]

[ \text{Chord Length} = 2 \times 11 \sin\left(\frac{7\pi}{8}\right) ]

[ \text{Chord Length} \approx 2 \times 11 \times 0.9239 ]

[ \text{Chord Length} \approx 20.465 ]

So, the length of the chord is approximately (20.465) 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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