I posted a problem "How Much Can a Goat Eat #2". Using a false assumption I calculated the overlapped area incorrectly. I request assistance in this regard. Please calculate the area of the green colored figure in this image. Thanks.

It looks like there are several ways to go about this, but this is my initial thought. First, we have \(AB = 3\sqrt{3}\). Thus \(\theta = \angle CAD = \cos^{-1}(\frac{\sqrt{3}}{4}) - \frac{\pi}{6}\). The area of the green region will then be the area of sector \(CAF\) minus the area of \(\Delta CAD\). This comes out to

Thanks, Brian. I obtained the same answer by using a different method.
In the sector ACF I calculated A by Law of Sines to be 34.34109373 degrees, altitude from AC to D to be 1.692355197.

Area of the green = Area of the sector - Area of triangle ADC

Area of the green = (34.34.../360 * Pi * 6^2) - (1/2 * 6* 1.69..)

Area of the green = 10.788573 - 5.0770656 = 5.7115074

This amount was taken to the problem "How Much Can a Goat Eat #2" and a corrected answer was obtained.

it can be solved by using integration in a very easy manner .just shift the whole diagram on a xy plane and do the integration by writing the equation .

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TopNewestIt looks like there are several ways to go about this, but this is my initial thought. First, we have \(AB = 3\sqrt{3}\). Thus \(\theta = \angle CAD = \cos^{-1}(\frac{\sqrt{3}}{4}) - \frac{\pi}{6}\). The area of the green region will then be the area of sector \(CAF\) minus the area of \(\Delta CAD\). This comes out to

\((\frac{1}{2})(6^{2})\theta - (\frac{1}{2})(3)(6\sin(\theta)) = 18\theta - 9\sin(\theta)\).

This "simplifies" to

\(18\cos^{-1}(\frac{\sqrt{3}}{4}) - 3\pi - (\frac{9}{8})\sqrt{3}(\sqrt{13} - 1)\),

which equals \(5.7115\) to \(4\) decimal places.

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Thanks, Brian. I obtained the same answer by using a different method. In the sector ACF I calculated A by Law of Sines to be 34.34109373 degrees, altitude from AC to D to be 1.692355197.

Area of the green = Area of the sector - Area of triangle ADC

Area of the green = (34.34.../360 * Pi * 6^2) - (1/2 * 6* 1.69..)

Area of the green = 10.788573 - 5.0770656 = 5.7115074

This amount was taken to the problem "How Much Can a Goat Eat #2" and a corrected answer was obtained.

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Glad we came to the same conclusion. I enjoyed the goat problem; thanks for posting it. :)

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yup its a good sum

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wow

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5.711

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Yup got 5.711

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49

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4.92 square units

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The area is 8.58

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How did you get that answer?

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it can be solved by using integration in a very easy manner .just shift the whole diagram on a xy plane and do the integration by writing the equation .

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5.748

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5.7115 Where did you err?

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what is the correct answer of this problem ????

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Look back at what Brian and I posted on Oct. 20.

If you think that you have the correct answer please submit your calculations.

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as i tried to solve it i got 4.9 as area of shaded region

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