SOLUTION: A man has a barn that is 20 ft by 10 ft. He tethers a cow to one corner of the outside of the barn using a 50-ft rope. What is the total area that the cow is capable of grazing?

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Question 80367: A man has a barn that is 20 ft by 10 ft. He tethers a cow to one corner of the outside of the barn using a 50-ft rope.
What is the total area that the cow is capable of grazing? I'm new to the site so I'l work on making a picture.

Thank you and it would be nice if youd show me the steps.
Found 2 solutions by stanbon, tomlaidlaw:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
The area of the circle the cow could cover if the barn was not there
is (pi)50^2
But the barn cuts of 1/4th of the circle leaving 3/4 th of the circle
for the cow to graze.
Grazing area = (3/4)(pi)50^2 = 5890.49 sq ft
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Cheers,
Stan H.

Answer by tomlaidlaw(14) About Me  (Show Source):
You can put this solution on YOUR website!
After the 3/4 circle at every corner the rope shortens by the length of the previous side, so the next portion of circle will not be as large. So you have 3/4 of a 50 radius circle + a sector of a 40 foot circle (blue) + a sector of a 30 foot circle (red) + a triangle with sides of 40, 22.36, and 30. minus half the area of the barn. The triangle is the overlap area of the two arcs, and its area can be figured by Heron's formula: A=sqrt%28s%28s-a%29%28s-b%29%28s-c%29%29 where s=%28a%2Bb%2Bc%29%2F2. I'll leave you to work the numbers. You'll also need the Law of Cosines to get the angles.
Tom laidlaw