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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
2 years, 5 months ago

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yup its a good sum Anunoy Chakraborty · 2 years, 5 months 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 · 2 years, 5 months ago

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@Brian Charlesworth 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 · 2 years, 5 months ago

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@Guiseppi Butel Glad we came to the same conclusion. I enjoyed the goat problem; thanks for posting it. :) Brian Charlesworth · 2 years, 5 months ago

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as i tried to solve it i got 4.9 as area of shaded region Deepali Shankar · 2 years, 4 months ago

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what is the correct answer of this problem ???? Deepali Shankar · 2 years, 4 months ago

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@Deepali Shankar 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 · 2 years, 4 months ago

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5.748 Mithun Mehedi Hasan · 2 years, 4 months ago

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@Mithun Mehedi Hasan 5.7115 Where did you err? Guiseppi Butel · 2 years, 4 months 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 · 2 years, 4 months ago

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The area is 8.58 Mohammed El-Sayed · 2 years, 4 months ago

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@Mohammed El-Sayed How did you get that answer? Guiseppi Butel · 2 years, 4 months ago

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4.92 square units Tony Sharma · 2 years, 4 months ago

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49 Rohit Dhiman · 2 years, 4 months ago

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Yup got 5.711 Mohit Ahuja · 2 years, 5 months ago

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5.711 Amardeep Singh · 2 years, 5 months ago

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wow Alleria Windrunner Gomez · 2 years, 5 months ago

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