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.

#Geometry

Easy Math Editor

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
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Comments

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 - 6 years, 7 months ago

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 - 6 years, 7 months ago

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

Brian Charlesworth - 6 years, 7 months ago

yup its a good sum

Anunoy Chakraborty - 6 years, 7 months ago

wow

Alleria Windrunner Gomez - 6 years, 7 months ago

5.711

Amardeep Singh - 6 years, 7 months ago

Yup got 5.711

Mohit Ahuja - 6 years, 7 months ago

Rohit Dhiman - 6 years, 7 months ago

4.92 square units

Tony Sharma - 6 years, 7 months ago

The area is 8.58

Mohammed El-Sayed - 6 years, 7 months ago

How did you get that answer?

Guiseppi Butel - 6 years, 7 months ago

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 - 6 years, 7 months ago

5.748

Mithun Mehedi Hasan - 6 years, 7 months ago

5.7115 Where did you err?

Guiseppi Butel - 6 years, 7 months ago

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

Deepali shankar - 6 years, 6 months ago

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 - 6 years, 6 months ago

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

Deepali shankar - 6 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$