A cow is tethered by a $50$m rope to a point $10$m from the corner of a $60$m-by-$30$m barn. What area of grass can be grazed?

Question

A cow is tethered by a 50-meter rope to a rectangular barn. The dimensions of the barn are 60 m x 30 m. The rope is fastened to a hook that is 10 meters from the corner on the longest side of the barn. Over exactly how much ground can the cow graze? (Assume that the cow cannot pass through the barn. He must graze outside only.)

I have drawn this problem as follows (not drawn to scale):

I am not even sure if the picture is correct, or how to apply any formula knowledge into this problem. In your explanation, please explain how and why you wrote the steps to your solution.

Answer 1

If the cow is outside of the barn then the area is consist of the area a semi circle and a quarter circle.

and if the cow is inside the barn then:

Answer 2

*not to scale

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If there is no restriction on the rope and cow can move freely outside the barn, then total area would be area of green cirlce(radius 50m) - area of barn inside the circle

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Hint : calculate the area of rectangle inside circle by dividing it into 3 parts.
area of rectangle inside circle as seen in image is A + B + C.
A = 30*10 (Area of rectangle) as cow can reach upper left corner of A, so it can graze it whole.
B = 1/2 * 30 * 40 (Area of Triangle) (height of triangle can be found using Pythagoras)

C = (subtended angle/360)*pi*50*50 (area of arc)
subtended angle = tanInverse(30/40)

300 + 600 + 803.37 = 1704.37 meter sq.

grazing area = pi*2500 - 1704.37 = 6149.61 meter sq.