Points #(2 ,4 )# and #(4 ,9 )# are #(3 pi)/4 # radians apart on a circle. What is the shortest arc length between the points?
Shortest Arc length
Chord radius Shortest Arc length
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To find the shortest arc length between two points on a circle, you can use the formula:
Arc Length = r * θ
Where:
- r is the radius of the circle.
- θ is the angle between the two points in radians.
First, we need to find the radius of the circle. We can use the distance formula to find the distance between the two points:
d = √((x2 - x1)^2 + (y2 - y1)^2)
Given the points (2, 4) and (4, 9), we have:
- x1 = 2, y1 = 4
- x2 = 4, y2 = 9
Substituting the values: d = √((4 - 2)^2 + (9 - 4)^2) = √(2^2 + 5^2) = √(4 + 25) = √29
Now, to find the angle between the points, we use the arctan formula:
θ = arctan((y2 - y1) / (x2 - x1))
Substituting the values: θ = arctan((9 - 4) / (4 - 2)) = arctan(5 / 2)
Using a calculator, arctan(5/2) ≈ 1.19029 radians
Now, we can calculate the arc length using the formula: Arc Length = r * θ
Since the radius is √29 and θ is approximately 1.19029 radians: Arc Length ≈ √29 * 1.19029 ≈ 4.664 units.
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To find the shortest arc length between two points on a circle, you can use the formula:
Arc Length = r * θ
where r is the radius of the circle and θ is the angle (in radians) subtended by the arc at the center of the circle.
First, find the radius of the circle using the distance formula between the given points:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d = sqrt((4 - 2)^2 + (9 - 4)^2) d = sqrt(2^2 + 5^2) d = sqrt(4 + 25) d = sqrt(29)
Now, you can find the angle between the points. Since the points are (3π/4) radians apart, the angle between them is 3π/4 radians.
Now, you can find the arc length:
Arc Length = r * θ Arc Length = sqrt(29) * (3π/4) Arc Length ≈ 3.741 * (3π/4) Arc Length ≈ 8.841
So, the shortest arc length between the points is approximately 8.841 units.
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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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