A chord with a length of #1 # runs from #pi/4 # to #pi/2 # radians on a circle. What is the area of the circle?
Area
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To find the area of the circle, we first need to determine the radius of the circle.
Given that the chord length is 1 and it subtends an arc from ( \frac{\pi}{4} ) to ( \frac{\pi}{2} ) radians, we can use the formula for the length of a chord in a circle in terms of the radius and the central angle:
[ \text{Chord Length} = 2r\sin\left(\frac{\theta}{2}\right) ]
Where ( r ) is the radius of the circle, ( \theta ) is the central angle subtended by the chord.
Given that the chord length is 1 and the central angle ( \theta = \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} ), we can solve for ( r ):
[ 1 = 2r\sin\left(\frac{\pi}{8}\right) ]
[ \sin\left(\frac{\pi}{8}\right) = \frac{1}{2r} ]
[ r = \frac{1}{2\sin\left(\frac{\pi}{8}\right)} ]
Now, we can find the area of the circle using the formula:
[ \text{Area of Circle} = \pi r^2 ]
Substitute the value of ( r ) to find the area of the circle.
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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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