# A circle has a chord that goes from #( pi)/4 # to #(13 pi) / 8 # radians on the circle. If the area of the circle is #32 pi #, what is the length of the chord?

Two radii and the chord form an isosceles triangle where "c" is the unknown length of the chord.

The lengths of the other two sides are:

The angle between the radii is:

We can use the Law of Cosines to find the length of the chord:

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To find the length of the chord, we first need to calculate the radius of the circle using the given area formula for a circle: ( A = \pi r^2 ). Given that the area of the circle is ( 32\pi ), we can solve for ( r ):

[ A = \pi r^2 ] [ 32\pi = \pi r^2 ]

Divide both sides by ( \pi ):

[ 32 = r^2 ]

Take the square root of both sides to find ( r ):

[ r = \sqrt{32} ]

[ r = 4\sqrt{2} ]

Now that we have the radius, we can use the central angle of the chord to find its length. The formula to calculate the length of the chord given the radius (( r )) and the central angle (( \theta )) is:

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

Given that the central angle ( \theta ) goes from ( \frac{\pi}{4} ) to ( \frac{13\pi}{8} ) radians, we can find ( \theta ) by subtracting the initial angle from the final angle:

[ \theta = \left(\frac{13\pi}{8}\right) - \left(\frac{\pi}{4}\right) ] [ \theta = \frac{13\pi - 2\pi}{8} ] [ \theta = \frac{11\pi}{8} ]

Now we can plug in the values into the chord length formula:

[ \text{Chord length} = 2(4\sqrt{2}) \sin\left(\frac{11\pi}{16}\right) ]

[ \text{Chord length} = 8\sqrt{2} \sin\left(\frac{11\pi}{16}\right) ]

[ \text{Chord length} \approx 8\sqrt{2} \times 0.923 ]

[ \text{Chord length} \approx 7.384 ]

So, the length of the chord is approximately ( 7.384 ) 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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