A circle's center is at #(3 ,1 )# and it passes through #(5 ,2 )#. What is the length of an arc covering #(7pi ) /12 # radians on the circle?
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To find the length of an arc covering ( \frac{7\pi}{12} ) radians on the circle, we first need to determine the radius of the circle.
The formula for the length of an arc in radians is ( \text{length} = r \times \text{angle} ), where ( r ) is the radius of the circle and the angle is in radians.
Given the center of the circle is at ( (3, 1) ) and it passes through ( (5, 2) ), we can use the distance formula to find the radius:
[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]
[ r = \sqrt{(5 - 3)^2 + (2 - 1)^2} ]
[ r = \sqrt{2^2 + 1^2} ]
[ r = \sqrt{4 + 1} ]
[ r = \sqrt{5} ]
So, the radius of the circle is ( \sqrt{5} ).
Now, to find the length of the arc covering ( \frac{7\pi}{12} ) radians:
[ \text{length} = r \times \text{angle} ]
[ \text{length} = \sqrt{5} \times \frac{7\pi}{12} ]
[ \text{length} = \frac{7\pi\sqrt{5}}{12} ]
Therefore, the length of the arc covering ( \frac{7\pi}{12} ) radians on the circle is ( \frac{7\pi\sqrt{5}}{12} ).
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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.
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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