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
5 years 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=33AB = 3\sqrt{3}. Thus θ=CAD=cos1(34)π6\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 CAFCAF minus the area of ΔCAD\Delta CAD. This comes out to

(12)(62)θ(12)(3)(6sin(θ))=18θ9sin(θ)(\frac{1}{2})(6^{2})\theta - (\frac{1}{2})(3)(6\sin(\theta)) = 18\theta - 9\sin(\theta).

This "simplifies" to

18cos1(34)3π(98)3(131)18\cos^{-1}(\frac{\sqrt{3}}{4}) - 3\pi - (\frac{9}{8})\sqrt{3}(\sqrt{13} - 1),

which equals 5.71155.7115 to 44 decimal places.

Brian Charlesworth - 5 years 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 - 5 years ago

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

Brian Charlesworth - 5 years ago

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yup its a good sum

Anunoy Chakraborty - 4 years, 12 months ago

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wow

Alleria Windrunner Gomez - 4 years, 11 months ago

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5.711

Amardeep Singh - 4 years, 11 months ago

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Yup got 5.711

Mohit Ahuja - 4 years, 11 months ago

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49

Rohit Dhiman - 4 years, 11 months ago

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4.92 square units

Tony Sharma - 4 years, 11 months ago

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The area is 8.58

Mohammed El-Sayed - 4 years, 11 months ago

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How did you get that answer?

Guiseppi Butel - 4 years, 11 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 - 4 years, 11 months ago

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5.748

Mithun Mehedi Hasan - 4 years, 11 months ago

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5.7115 Where did you err?

Guiseppi Butel - 4 years, 11 months ago

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

Deepali shankar - 4 years, 10 months 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 - 4 years, 10 months ago

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

Deepali shankar - 4 years, 10 months ago

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