# A circle has a chord that goes from #pi/3 # to #pi/8 # radians on the circle. If the area of the circle is #25 pi #, what is the length of the chord?

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To find the length of the chord, we can use the formula:

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

Where:

- ( r ) is the radius of the circle.
- ( \theta ) is the angle subtended by the chord at the center of the circle.

Given:

- ( \theta = \frac{\pi}{3} - \frac{\pi}{8} )
- The area of the circle is ( 25\pi )

We know that the area of a circle is given by the formula ( A = \pi r^2 ). So, if the area of the circle is ( 25\pi ), then:

[ \pi r^2 = 25\pi ] [ r^2 = 25 ] [ r = 5 ]

Now, we can substitute the values of ( r ) and ( \theta ) into the formula for the chord length:

[ \text{Chord Length} = 2(5) \sin\left(\frac{\frac{\pi}{3} - \frac{\pi}{8}}{2}\right) ]

[ \text{Chord Length} = 10 \sin\left(\frac{\frac{\pi}{3} - \frac{\pi}{8}}{2}\right) ]

[ \text{Chord Length} = 10 \sin\left(\frac{\frac{5\pi}{24}}{2}\right) ]

[ \text{Chord Length} = 10 \sin\left(\frac{5\pi}{48}\right) ]

Now, using a calculator to find the value of ( \sin\left(\frac{5\pi}{48}\right) ), and then multiplying it by 10, we can determine the length of the chord.

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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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