# A circle's center is at #(3 ,4 )# and it passes through #(0 ,2 )#. What is the length of an arc covering #( pi ) /6 # radians on the circle?

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To find the length of an arc covering ( \frac{\pi}{6} ) radians on the circle, we first need to calculate the radius of the circle using the given center and a point on the circle.

The formula to find the distance between two points ((x_1, y_1)) and ((x_2, y_2)) is:

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

Given that the center of the circle is at ((3, 4)) and it passes through ((0, 2)), we can use these coordinates to find the radius.

[ \text{Radius} = \sqrt{(0 - 3)^2 + (2 - 4)^2} ] [ \text{Radius} = \sqrt{(-3)^2 + (-2)^2} ] [ \text{Radius} = \sqrt{9 + 4} ] [ \text{Radius} = \sqrt{13} ]

Now that we have the radius (( \sqrt{13} )), we can use the formula for the length of an arc on a circle:

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

Substituting the given angle of ( \frac{\pi}{6} ) radians:

[ \text{Arc Length} = \sqrt{13} \times \frac{\pi}{6} ]

[ \text{Arc Length} = \frac{\sqrt{13}\pi}{6} ]

Therefore, the length of the arc covering ( \frac{\pi}{6} ) radians on the circle is ( \frac{\sqrt{13}\pi}{6} ) units.

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To find the length of an arc covering ( \frac{\pi}{6} ) radians on the circle, we first need to find the radius of the circle using the given information about its center and a point it passes through.

Using the distance formula, the distance between the center of the circle at (3, 4) and the point (0, 2) on the circle can be calculated as:

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

[ r = \sqrt{(0 - 3)^2 + (2 - 4)^2} ]

[ r = \sqrt{9 + 4} ]

[ r = \sqrt{13} ]

Now that we have the radius (( \sqrt{13} )), we can use the formula for the length of an arc of a circle:

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

[ \text{Arc Length} = \sqrt{13} \times \frac{\pi}{6} ]

[ \text{Arc Length} = \frac{\sqrt{13}\pi}{6} ]

So, the length of an arc covering ( \frac{\pi}{6} ) radians on the circle is ( \frac{\sqrt{13}\pi}{6} ) units.

By signing up, you agree to our Terms of Service and Privacy Policy

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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