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A rectangular building is 4 meters by 6 meters. A dog is tied to a rope that is 10 meters long, and the other end is tied to the midpoint of one of the short sides of the building. Find the total area of the region that the dog can reach (not including the inside of the building), in square meters. Plz don't send me the link of the simlilar problem, since I don't understand it. 

 Feb 17, 2024
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ok ill make thise easy to understand so:

we can find the area the dog can reach on one side then multiply by 2 to get the total area. Imagine wrapping a string around a sqaure. If we start wrapping the 10m string around the building towards the right then the string will form a quarter circle till the side of the building and the pivot will change to the corner of the building and the radius that makes the next semicircle will be \(10-2=8\). Like wise this will continue to the side of the building the the pivot will again change to be the far corner of the building. Then the radius will change again to \(8-6=2\). Finally there will be another quarter circle with radius 2. There is no overlap as the other side will copy the same area due to symmetry. 

So:
\(2(\frac{2^2\pi}{4}+\frac{8^2\pi}{4}+\frac{10^2\pi}{4})=\boxed{84\pi\hspace{2mm}\text{ m }^2}\)

 Feb 17, 2024

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