# A circle has a chord that goes from #( 2 pi)/3 # to #(11 pi) / 12 # radians on the circle. If the area of the circle is #5 pi #, what is the length of the chord?

Length of the chord

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To find the length of the chord in a circle with a given area, you can use the following formula:

[ \text{Area of the circle} = \frac{1}{2} r^2 \theta ]

Where:

- ( r ) is the radius of the circle
- ( \theta ) is the central angle in radians

Given: Area of the circle = ( 5\pi ) Central angle ( \theta = \frac{11\pi}{12} - \frac{2\pi}{3} )

First, find the radius ( r ) using the given area:

[ 5\pi = \frac{1}{2} r^2 \theta ] [ 5\pi = \frac{1}{2} r^2 \left( \frac{11\pi}{12} - \frac{2\pi}{3} \right) ]

Solve for ( r ):

[ 5\pi = \frac{1}{2} r^2 \left( \frac{11\pi}{12} - \frac{8\pi}{12} \right) ] [ 5\pi = \frac{1}{2} r^2 \times \frac{3\pi}{12} ] [ 5\pi = \frac{1}{8} r^2 \pi ]

Now, solve for ( r ):

[ r^2 = 40 ] [ r = \sqrt{40} ] [ r = 2\sqrt{10} ]

Next, calculate the length of the chord using the radius and central angle:

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

Substitute the values:

[ \text{Length of chord} = 2(2\sqrt{10}) \sin\left(\frac{\frac{11\pi}{12} - \frac{2\pi}{3}}{2}\right) ]

[ \text{Length of chord} = 4\sqrt{10} \sin\left(\frac{\frac{11\pi}{12} - \frac{8\pi}{12}}{2}\right) ] [ \text{Length of chord} = 4\sqrt{10} \sin\left(\frac{3\pi}{24}\right) ] [ \text{Length of chord} = 4\sqrt{10} \sin\left(\frac{\pi}{8}\right) ]

Using the sine value of ( \frac{\pi}{8} ) (approximately 0.3827):

[ \text{Length of chord} = 4\sqrt{10} \times 0.3827 ] [ \text{Length of chord} = 1.5308 \times 2\sqrt{10} ] [ \text{Length of chord} = 3.0616\sqrt{10} ] [ \text{Length of chord} \approx 3.0616 \times 3.1623 ] [ \text{Length of chord} \approx 9.683 ] (rounded to three decimal places)

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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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