A chord with a length of #12 # runs from #pi/3 # to #pi/2 # radians on a circle. What is the area of the circle?
Area of the circle is 1688.3371
Chord length = 12 Area of the circle =
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The area of the circle can be determined using the formula for the area of a sector.
Given:
- The length of the chord is 12 units.
- The chord subtends an angle of π/2 - π/3 = π/6 radians.
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Find the radius of the circle using the law of cosines: ( c^2 = a^2 + b^2 - 2ab \cos(C) ) where ( a ) and ( b ) are the radii, ( c ) is the length of the chord, and ( C ) is the angle subtended by the chord.
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Once you have the radius, use the formula for the area of a circle (( A = \pi r^2 )) to find the area.
Let's denote the radius of the circle as ( r ).
Using the law of cosines: [ 12^2 = r^2 + r^2 - 2rr \cos(\frac{\pi}{6}) ]
[ 144 = 2r^2 - 2r^2 \cos(\frac{\pi}{6}) ]
[ 144 = 2r^2 - r^2 \sqrt{3} ]
[ 144 = r^2(2 - \sqrt{3}) ]
[ r^2 = \frac{144}{2 - \sqrt{3}} ]
[ r^2 = \frac{144(2 + \sqrt{3})}{(2 - \sqrt{3})(2 + \sqrt{3})} ]
[ r^2 = \frac{144(2 + \sqrt{3})}{4 - 3} ]
[ r^2 = 144(2 + \sqrt{3}) ]
[ r = \sqrt{144(2 + \sqrt{3})} ]
Once you have the value of ( r ), you can find the area of the circle using the formula ( A = \pi r^2 ).
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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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