A chord with a length of #16 # runs from #pi/4 # to #pi/2 # radians on a circle. What is the area of the circle?
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To find the area of the circle, we need to know the radius.
The chord subtends an angle of ( \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4} ) radians at the center of the circle, which means the corresponding central angle is ( \frac{\pi}{4} ) radians.
In a circle, the chord is perpendicular to the radius at the midpoint of the chord. So, if we draw a radius from the center to the midpoint of the chord, it will bisect the chord. This forms a right triangle with the radius as the hypotenuse, and half of the chord length (since it's bisected) as one of the legs.
Using trigonometry, we can find the length of the radius:
[ \sin\left(\frac{\pi}{4}\right) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\frac{16}{2}}{r} ]
Solving for ( r ):
[ r = \frac{16}{2\sin\left(\frac{\pi}{4}\right)} = \frac{16}{2 \times \frac{\sqrt{2}}{2}} = \frac{16}{\sqrt{2}} = 8\sqrt{2} ]
Now that we have the radius, we can find the area of the circle:
[ \text{Area} = \pi r^2 = \pi \times (8\sqrt{2})^2 = 64\pi ]
So, the area of the circle is ( 64\pi ) square units.
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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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