# A circle's center is at #(9 ,4 )# and it passes through #(1 ,6 )#. What is the length of an arc covering #(2pi ) /3 # radians on the circle?

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To find the length of an arc covering ( \frac{2\pi}{3} ) radians on the circle, we need to calculate the radius of the circle first using the given coordinates of the center and a point on the circle. Then, we can use the formula for the arc length of a circle, which is ( s = r \times \theta ), where ( s ) is the arc length, ( r ) is the radius of the circle, and ( \theta ) is the angle in radians subtended by the arc.

Given: Center of the circle: ( (9, 4) ) Point on the circle: ( (1, 6) ) Angle ( \theta = \frac{2\pi}{3} )

Using the distance formula to find the radius: [ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Substituting the coordinates of the center and the point on the circle: [ r = \sqrt{(1 - 9)^2 + (6 - 4)^2} ] [ r = \sqrt{(-8)^2 + (2)^2} ] [ r = \sqrt{64 + 4} ] [ r = \sqrt{68} ] [ r = 2\sqrt{17} ]

Now, we can calculate the arc length using the formula: [ s = r \times \theta ] [ s = (2\sqrt{17}) \times \frac{2\pi}{3} ] [ s = \frac{4\pi\sqrt{17}}{3} ]

Therefore, the length of the arc covering ( \frac{2\pi}{3} ) radians on the circle is ( \frac{4\pi\sqrt{17}}{3} ) 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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