A circle's center is at #(3 ,1 )# and it passes through #(6 ,6 )#. What is the length of an arc covering #(7pi ) /12 # radians on the circle?
(WARNING: May be grossly worded, I will try to fix it later)
We know two points of the circle. We know the center (3,1), and we know a point the circle passes through (6,6). Since we know a point on the circle and the center, we can use them to find the radius, since a radius is any line on a circle that goes from the center to any point on the circle.
So we need to use the distance formula:
Now we have the radius. Our goal is still to find the circumference. To get the circumference with the radius, we use the formula for circumference:
Plug in radius:
Okay, now we reached our first goal, which was to find the circumference. Now we can move on to finding the arc length.
We now have a fraction to use on our circumference. We want to know the arc length that takes up 7 parts of 24, so we multiply 7/24 by the circumference, the circumference being the total arc length.
If you simplify that, you will get the answer:
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When the central angle
For this problem:
If we know the central angle, then we only need to find the radius of the circle, which is the distance from the center to any point 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 calculate the radius of the circle using the given center and a point on the circle.
Using the distance formula, the distance between the center ((3, 1)) and the point ((6, 6)) is:
[ \sqrt{(6 - 3)^2 + (6 - 1)^2} = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} ]
So, the radius of the circle is ( \sqrt{34} ).
The formula to find the length of an arc is ( s = r \cdot \theta ), where ( s ) is the arc length, ( r ) is the radius, and ( \theta ) is the angle in radians.
Thus, the length of the arc covering ( \frac{7\pi}{12} ) radians is:
[ s = \sqrt{34} \times \frac{7\pi}{12} = \frac{7\pi \sqrt{34}}{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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