# A circle has a chord that goes from #( 5 pi)/4 # to #(4 pi) / 3 # radians on the circle. If the area of the circle is #42 pi #, what is the length of the chord?

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To find the length of the chord, we need to use the formula for the length of a chord in a circle, which is (2r \sin(\frac{\theta}{2})), 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 (42\pi), we can use the formula for the area of a circle, (A = \pi r^2), to find the radius of the circle.

(42\pi = \pi r^2)

(r^2 = 42)

(r = \sqrt{42})

Now, we need to find the angle (\theta) subtended by the chord at the center of the circle. The chord spans from (\frac{5\pi}{4}) to (\frac{4\pi}{3}) radians.

To find the angle subtended by the chord, we subtract the smaller angle from the larger angle:

(\theta = \frac{4\pi}{3} - \frac{5\pi}{4})

(\theta = \frac{16\pi}{12} - \frac{15\pi}{12})

(\theta = \frac{\pi}{12})

Now, we can use the chord length formula:

(2r \sin\left(\frac{\theta}{2}\right) = 2\sqrt{42} \sin\left(\frac{\pi}{24}\right))

Using a calculator to evaluate (\sin\left(\frac{\pi}{24}\right)), we find:

(\sin\left(\frac{\pi}{24}\right) \approx 0.1305)

Substituting this value into the formula:

(2\sqrt{42} \times 0.1305 \approx 5.5003)

So, the length of the chord is approximately (5.5003) 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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