A circle has a chord that goes from #( 2 pi)/3 # to #(11 pi) / 12 # radians on the circle. If the area of the circle is #5 pi #, what is the length of the chord?

Answer 1

Length of the chord#=1.76#

A chord that goes from #(2pi)/3#to #(11pi)/12# so it travels the distance #(11pi)/12-(2pi)/3=(pi)/4#; or #(pi)/4-:2pi=1/8# of the Circumference of the Circle Area of the Circle#=pir^2=5pi# or #r^2=5# or #r=sqrt5# or #r=2.24# Circumference of the circle#=2pir=2(pi)(2.24)=14.05# Therefore Length of the chord#=14.05times1/8=1.76#
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Answer 2

To find the length of the chord in a circle with a given area, you can use the following formula:

[ \text{Area of the circle} = \frac{1}{2} r^2 \theta ]

Where:

  • ( r ) is the radius of the circle
  • ( \theta ) is the central angle in radians

Given: Area of the circle = ( 5\pi ) Central angle ( \theta = \frac{11\pi}{12} - \frac{2\pi}{3} )

First, find the radius ( r ) using the given area:

[ 5\pi = \frac{1}{2} r^2 \theta ] [ 5\pi = \frac{1}{2} r^2 \left( \frac{11\pi}{12} - \frac{2\pi}{3} \right) ]

Solve for ( r ):

[ 5\pi = \frac{1}{2} r^2 \left( \frac{11\pi}{12} - \frac{8\pi}{12} \right) ] [ 5\pi = \frac{1}{2} r^2 \times \frac{3\pi}{12} ] [ 5\pi = \frac{1}{8} r^2 \pi ]

Now, solve for ( r ):

[ r^2 = 40 ] [ r = \sqrt{40} ] [ r = 2\sqrt{10} ]

Next, calculate the length of the chord using the radius and central angle:

[ \text{Length of chord} = 2r \sin\left(\frac{\theta}{2}\right) ]

Substitute the values:

[ \text{Length of chord} = 2(2\sqrt{10}) \sin\left(\frac{\frac{11\pi}{12} - \frac{2\pi}{3}}{2}\right) ]

[ \text{Length of chord} = 4\sqrt{10} \sin\left(\frac{\frac{11\pi}{12} - \frac{8\pi}{12}}{2}\right) ] [ \text{Length of chord} = 4\sqrt{10} \sin\left(\frac{3\pi}{24}\right) ] [ \text{Length of chord} = 4\sqrt{10} \sin\left(\frac{\pi}{8}\right) ]

Using the sine value of ( \frac{\pi}{8} ) (approximately 0.3827):

[ \text{Length of chord} = 4\sqrt{10} \times 0.3827 ] [ \text{Length of chord} = 1.5308 \times 2\sqrt{10} ] [ \text{Length of chord} = 3.0616\sqrt{10} ] [ \text{Length of chord} \approx 3.0616 \times 3.1623 ] [ \text{Length of chord} \approx 9.683 ] (rounded to three decimal places)

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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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