# A circle's center is at #(1 ,2 )# and it passes through #(5 ,7 )#. What is the length of an arc covering #(5pi ) /3 # radians on the circle?

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To find the length of an arc covering ( \frac{5\pi}{3} ) radians on the circle, you need to first find the radius of the circle using the distance formula. Then, you can use the formula for the arc length of a circle, which is given by ( s = r \cdot \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 the center of the circle at (1, 2) and it passes through (5, 7), you can use the distance formula to find the radius.

The distance formula is ( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ), where ( (x_1, y_1) ) and ( (x_2, y_2) ) are the coordinates of two points.

So, using the distance formula with the center (1, 2) and the point (5, 7), you get:

( d = \sqrt{(5 - 1)^2 + (7 - 2)^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} ).

Now that you have the radius, you can use the formula for arc length:

( s = r \cdot \theta ).

Substituting the values, you get:

( s = \sqrt{41} \cdot \frac{5\pi}{3} ).

So, the length of the arc covering ( \frac{5\pi}{3} ) radians on the circle is ( \sqrt{41} \cdot \frac{5\pi}{3} ).

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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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