# A circle has a chord that goes from #( pi)/3 # to #(7 pi) / 12 # radians on the circle. If the area of the circle is #16 pi #, what is the length of the chord?

The chord length is

Use the equation for the area of the circle to find the radius:

Compute the angle:

Because the chord and two radii form a triangle, one can use the Law of Cosines to find the length of the chord, c:

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To find the length of the chord, we first need to determine the radius of the circle. The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.

Given that the area of the circle is 16π, we can set up the equation as follows: 16π = πr^2

Dividing both sides by π: 16 = r^2

Taking the square root of both sides: r = 4

Now that we have the radius of the circle, we can use it to find the length of the chord. The formula to calculate the length of a chord in a circle using the central angle (in radians) is: Length of chord = 2 * radius * sin(θ/2)

Given that the central angle θ goes from π/3 to 7π/12 radians, we can find the length of the chord as follows:

θ = (7π/12) - (π/3) = (7π - 4π)/12 = (3π)/12 = π/4

Substituting the values into the formula: Length of chord = 2 * 4 * sin(π/8)

Using the half-angle formula for sine: sin(π/8) = √((1 - cos(π/4)) / 2)

cos(π/4) = √2 / 2

Substituting the value: sin(π/8) = √((1 - √2/2) / 2)

Now, we can find the length of the chord: Length of chord = 2 * 4 * √((1 - √2/2) / 2)

Length of chord ≈ 3.03 unitsTo find the length of the chord, you can use the formula:

[ \text{Chord Length} = 2 \times \text{radius} \times \sin \left( \frac{\text{central angle}}{2} \right) ]

Given that the area of the circle is (16\pi), you can use the formula for the area of a circle to find the radius:

[ \text{Area of Circle} = \pi \times \text{radius}^2 ]

Then solve for the radius:

[ \text{radius}^2 = \frac{\text{Area of Circle}}{\pi} ] [ \text{radius}^2 = \frac{16\pi}{\pi} ] [ \text{radius}^2 = 16 ]

[ \text{radius} = 4 ]

Now, you have the radius of the circle, and you know the central angle of the chord is (\frac{7\pi}{12} - \frac{\pi}{3} = \frac{7\pi - 4\pi}{12} = \frac{3\pi}{12} = \frac{\pi}{4}). Plug these values into the formula for the length of the chord:

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

[ \text{Chord Length} = 8 \times \sin \left( \frac{\pi}{4} \right) ]

Using the fact that (\sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}), you get:

[ \text{Chord Length} = 8 \times \frac{\sqrt{2}}{2} ]

[ \text{Chord Length} = 8\sqrt{2} ]

So, the length of the chord is (8\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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