# 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 #48 pi #, what is the length of the chord?

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To find the length of the chord, you 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.

First, we need to find the radius of the circle using the given area:

[ A = \pi r^2 ]

[ 48\pi = \pi r^2 ]

[ r^2 = 48 ]

[ r = \sqrt{48} = 4\sqrt{3} ]

Now, we need to find the length of the chord using the formula:

[ \text{Chord length} = 2(4\sqrt{3}) \sin\left(\frac{15\pi/8 - \pi/2}{2}\right) ]

[ = 8\sqrt{3} \sin\left(\frac{15\pi/8 - \pi/2}{2}\right) ]

[ = 8\sqrt{3} \sin\left(\frac{15\pi/8 - 4\pi/8}{2}\right) ]

[ = 8\sqrt{3} \sin\left(\frac{11\pi}{16}\right) ]

Now, calculate ( \sin\left(\frac{11\pi}{16}\right) ), then multiply by ( 8\sqrt{3} ) to find 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.

- A triangle has corners at #(2 , 2 )#, #(1 ,3 )#, and #(6 ,4 )#. What is the radius of the triangle's inscribed circle?
- A triangle has corners at #(4 ,7 )#, #(1 ,3 )#, and #(6 ,5 )#. What is the area of the triangle's circumscribed circle?
- A circle has a center at #(3 ,0 )# and passes through #(0 ,1 )#. What is the length of an arc covering #(3pi ) /4 # radians on the circle?
- A circle's center is at #(2 ,1 )# and it passes through #(0 ,7 )#. What is the length of an arc covering #(5pi ) /12 # radians on the circle?
- A triangle has vertices A, B, and C. Vertex A has an angle of #pi/8 #, vertex B has an angle of #(pi)/12 #, and the triangle's area is #6 #. What is the area of the triangle's incircle?

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