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 #27 pi #, what is the length of the chord?
Length of the chord
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To find the length of the chord, you can use the formula for the area of a circle and the properties of chords in circles.

The area of a circle is given by ( A = \pi r^2 ), where ( A ) is the area and ( r ) is the radius of the circle.

Given that the area of the circle is ( 27 \pi ), you can solve for the radius: ( 27 \pi = \pi r^2 ) ( r^2 = 27 ) ( r = \sqrt{27} = 3\sqrt{3} )

Now, to find the length of the chord, you can use the formula for the length of a chord in a circle: ( \text{Chord Length} = 2r \sin\left(\frac{\theta}{2}\right) ), where ( \theta ) is the angle subtended by the chord at the center of the circle.

The angle subtended by the chord is ( \frac{13\pi}{8}  \frac{\pi}{4} = \frac{13\pi  2\pi}{8} = \frac{11\pi}{8} ).

Substitute the values into the formula: ( \text{Chord Length} = 2 \times 3\sqrt{3} \sin\left(\frac{11\pi}{16}\right) )

Calculate the length: ( \text{Chord Length} \approx 2 \times 3\sqrt{3} \times 0.83147 ) (approximately) ( \text{Chord Length} \approx 4.989 \times \sqrt{3} )
So, the length of the chord is approximately ( 4.989 \times \sqrt{3} ) units.
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