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Finding Area

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.

Note by Guiseppi Butel
3 years, 1 month ago

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yup its a good sum

Anunoy Chakraborty - 3 years, 1 month ago

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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

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

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.

Brian Charlesworth - 3 years, 1 month ago

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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.

Guiseppi Butel - 3 years, 1 month ago

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

Brian Charlesworth - 3 years, 1 month ago

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

Deepali Shankar - 3 years ago

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

Deepali Shankar - 3 years ago

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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.

Guiseppi Butel - 3 years ago

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5.748

Mithun Mehedi Hasan - 3 years, 1 month ago

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5.7115 Where did you err?

Guiseppi Butel - 3 years, 1 month ago

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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 .

Shubham Kumar Sah Sah - 3 years, 1 month ago

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The area is 8.58

Mohammed El-Sayed - 3 years, 1 month ago

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How did you get that answer?

Guiseppi Butel - 3 years, 1 month ago

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4.92 square units

Tony Sharma - 3 years, 1 month ago

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49

Rohit Dhiman - 3 years, 1 month ago

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Yup got 5.711

Mohit Ahuja - 3 years, 1 month ago

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5.711

Amardeep Singh - 3 years, 1 month ago

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wow

Alleria Windrunner Gomez - 3 years, 1 month ago

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