# A circle's center is at #(2 ,4 )# and it passes through #(7 ,6 )#. What is the length of an arc covering #(15pi ) /8 # radians on the circle?

Length of the arc:

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To find the length of an arc covering ( \frac{15\pi}{8} ) radians on the circle, we need to first find the radius of the circle using the given points.

Given:

- Center of the circle: ( (2, 4) )
- Point on the circle: ( (7, 6) )

Using the distance formula between two points, we can find the radius:

[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Substituting the given points:

[ \text{Distance} = \sqrt{(7 - 2)^2 + (6 - 4)^2} ]

[ = \sqrt{5^2 + 2^2} ]

[ = \sqrt{25 + 4} ]

[ = \sqrt{29} ]

Now, the length of an arc on a circle is given by:

[ \text{Length of Arc} = r \times \text{angle in radians} ]

Given:

- ( \text{Angle} = \frac{15\pi}{8} )
- ( r = \sqrt{29} )

[ \text{Length of Arc} = \sqrt{29} \times \frac{15\pi}{8} ]

[ = \frac{15\pi \sqrt{29}}{8} ]

So, the length of an arc covering ( \frac{15\pi}{8} ) radians on the circle is ( \frac{15\pi \sqrt{29}}{8} ).

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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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