Points #(6 ,7 )# and #(7 ,5 )# are #(2 pi)/3 # radians apart on a circle. What is the shortest arc length between the points?

Answer 1

#2/9*sqrt(15)*pi~=2.704#

Refer to the figure below

#alpha=(2pi)/3 radians#
Or
#alpha=(2*180^@)/3=120^@#

#AB=sqrt((7-6)^2+(5-7)^2)=sqrt(1+4)=sqrt(5)#

Applying Law of Cosines in #triangle_(ABC)#:

#AB^2=r^2+r^2-2r*r*cos 180^@#
#2r^2-2r^2*(-1/2)=(sqrt(5))^2#
#3r^2=5# => #r=sqrt(5/3)#

Length of the arc

# arc AB=r*alpha#, where alpha is given in radians
#arc AB=sqrt(5/3)*(2*pi)/3*(sqrt(3)/sqrt(3))=2/9*sqrt(15)*pi#

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Answer 2

To find the shortest arc length between two points on a circle, you can use the formula for arc length (s) given by:

[s = r \cdot \theta]

Where (r) is the radius of the circle and (\theta) is the central angle between the two points measured in radians.

Given that the points are ((6, 7)) and ((7, 5)) and they are (\frac{2\pi}{3}) radians apart, we first find the radius of the circle using the distance formula:

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

Plugging in the coordinates, we get:

[d = \sqrt{(7 - 6)^2 + (5 - 7)^2} = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}]

Now, we use the formula for arc length:

[s = r \cdot \theta]

[s = \sqrt{5} \cdot \frac{2\pi}{3} = \frac{2\pi\sqrt{5}}{3}]

So, the shortest arc length between the points is (\frac{2\pi\sqrt{5}}{3}) units.

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Answer from HIX Tutor

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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