A circle has a chord that goes from #( pi)/2 # to #(15 pi) / 8 # radians on the circle. If the area of the circle is #48 pi #, what is the length of the chord?

Answer 1

#color(green)("Chord Length " d ~~ 12.8#

#"Given : " delta = (15pi)/8 - pi / 2 = (11pi) / 8 #

#"Area of the circle " A = pi R^2 = 48 pi#

#R = (sqrt 48 *cancel pi ) /cancel pi = sqrt 48 = 4 sqrt3#

#"Chord Length " d = 2 * R sin delta = 2 * 4 sqrt3 * sin ((11pi)/8)#

#d = 8 sqrt 3 * sin ((11pi) / 8) ~~ 12.8#

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Answer 2

To find the length of the chord, you can use the formula:

[ \text{Chord length} = 2r \sin\left(\frac{\theta}{2}\right) ]

Where:

  • ( r ) is the radius of the circle.
  • ( \theta ) is the angle subtended by the chord at the center of the circle.

First, we need to find the radius of the circle using the given area:

[ A = \pi r^2 ]

[ 48\pi = \pi r^2 ]

[ r^2 = 48 ]

[ r = \sqrt{48} = 4\sqrt{3} ]

Now, we need to find the length of the chord using the formula:

[ \text{Chord length} = 2(4\sqrt{3}) \sin\left(\frac{15\pi/8 - \pi/2}{2}\right) ]

[ = 8\sqrt{3} \sin\left(\frac{15\pi/8 - \pi/2}{2}\right) ]

[ = 8\sqrt{3} \sin\left(\frac{15\pi/8 - 4\pi/8}{2}\right) ]

[ = 8\sqrt{3} \sin\left(\frac{11\pi}{16}\right) ]

Now, calculate ( \sin\left(\frac{11\pi}{16}\right) ), then multiply by ( 8\sqrt{3} ) to find the length of the chord.

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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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