# A circle has a chord that goes from #( 3 pi)/4 # to #(5 pi) / 4 # radians on the circle. If the area of the circle is #144 pi #, what is the length of the chord?

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The length of the chord in a circle can be calculated using the formula:

[ \text{Length of chord} = 2 \times \text{radius} \times \sin\left(\frac{\theta}{2}\right) ]

Where:

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

Given that the area of the circle is ( 144\pi ), we can use the formula for the area of a circle to find the radius (( r )).

[ \text{Area of circle} = \pi \times r^2 ] [ 144\pi = \pi \times r^2 ]

From this, we find that ( r = 12 ).

Now, we can calculate the length of the chord using the given angles ( \frac{3\pi}{4} ) and ( \frac{5\pi}{4} ).

[ \theta = \frac{5\pi}{4} - \frac{3\pi}{4} = \frac{\pi}{2} ]

[ \text{Length of chord} = 2 \times 12 \times \sin\left(\frac{\pi}{4}\right) ]

[ = 2 \times 12 \times \frac{\sqrt{2}}{2} ]

[ = 12 \sqrt{2} ]

So, the length of the chord is ( 12\sqrt{2} ) 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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