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?

Answer 1

The area of the circle is #6753.36#

The chord forms the base of of an isosceles triangle where the legs are the radii

The internal angles of a triangle sum to #Pi# The base angles of an isosceles triangle, where the apex is #a# radians can be calculated as #(Pi - a)/2#
The angle at the apex of the triangle is #Pi/6# Therefore, the equal angles are #(Pi - Pi/6)/2# or #5Pi/12#radians

The leg of an isosceles triangle can be calculated as half the base divided by the cosine of the base angle

Therefore, the radius (#r#) can now be calculated as #r = 24/2 * 1/cos(5Pi/12) = 46.36444#
The area of a circle is given by #Pir^2#
The area of the circle is #Pi*46.36444*46.36444 = 6753.36#
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Answer 2

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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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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