A chord with a length of #24 # runs from #pi/3 # to #pi/2 # radians on a circle. What is the area of the circle?
The area of the circle is
The chord forms the base of of an isosceles triangle where the legs are the radii
The leg of an isosceles triangle can be calculated as half the base divided by the cosine of the base angle
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To find the area of the circle, we first need to find the radius. We can use the formula for the length of a chord in a circle, which is ( L = 2r \sin(\frac{\theta}{2}) ), where ( L ) is the length of the chord, ( r ) is the radius of the circle, and ( \theta ) is the central angle in radians.
Given that the length of the chord is 24 units and the central angle is ( \frac{\pi}{2} - \frac{\pi}{3} = \frac{\pi}{6} ) radians, we can rearrange the formula to solve for the radius:
[ r = \frac{L}{2 \sin(\frac{\theta}{2})} ]
Plugging in the values:
[ r = \frac{24}{2 \sin(\frac{\pi}{12})} ]
[ r = \frac{24}{2 \sin(\frac{\pi}{12})} ]
[ r ≈ \frac{24}{2 \times 0.2588} ]
[ r ≈ \frac{24}{0.5176} ]
[ r ≈ 46.36 ]
Now that we have the radius, we can use the formula for the area of a circle, which is ( A = \pi r^2 ), where ( A ) is the area of the circle and ( r ) is the radius.
Plugging in the radius we found:
[ A = \pi (46.36)^2 ]
[ A ≈ \pi \times 2152.06 ]
[ A ≈ 6766.29 , \text{square units} ]
So, the area of the circle is approximately 6766.29 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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