# A circle has a center at #(3 ,0 )# and passes through #(1 ,1 )#. What is the length of an arc covering #(3pi ) /4 # radians on the circle?

The length of the arc is

The radius of the circle is

The length of the arc is

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To find the length of an arc covering ( \frac{3\pi}{4} ) radians on the circle, we use the formula for the length of an arc on a circle:

[ \text{Arc Length} = \text{Radius} \times \text{Angle (in radians)} ]

First, we need to find the radius of the circle. We can use the distance formula to find the distance between the center of the circle at (3, 0) and a point on the circle at (1, 1).

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

[ \text{Distance} = \sqrt{(1 - 3)^2 + (1 - 0)^2} ]

[ \text{Distance} = \sqrt{(-2)^2 + (1)^2} ]

[ \text{Distance} = \sqrt{4 + 1} ]

[ \text{Distance} = \sqrt{5} ]

So, the radius of the circle is ( \sqrt{5} ).

Now, we can find the length of the arc covering ( \frac{3\pi}{4} ) radians:

[ \text{Arc Length} = \sqrt{5} \times \frac{3\pi}{4} ]

[ \text{Arc Length} = \frac{3\pi\sqrt{5}}{4} ]

Therefore, the length of the arc covering ( \frac{3\pi}{4} ) radians on the circle is ( \frac{3\pi\sqrt{5}}{4} ).

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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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