# 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?

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.

• In your drawing the tether is attached to the inside of the barn. I would assume the hook was on the outside of the barn. – Jens Oct 22 '17 at 18:51
• As @Jens suggests, it makes more sense for the hook to be on the outside of the barn. Otherwise, the rope gets from inside the barn to outside it through some kind of opening, and we'd need information about the location and size of the opening. – Blue Oct 22 '17 at 19:29
• Related: Goat tethered by $8$m rope to corner of $4\times 6$ barn. Also, $6$m rope, $3\times 4$ house. Or even Two goats (although the special premise of this question ---that the goats avoid grass they can both reach--- doesn't seem to come into play). And other variants, including circular fields. Just search for "graze", "grass", "goat", "cow", "sheep", etc. – Blue Oct 22 '17 at 19:34

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:

• Thank you for your detailed explanation, @Seyed. Your first visual is more likely the solution to the problem. I do have a question: why would the area consist of a half circle/semi-circle? – Bibliophile Oct 25 '17 at 10:06
• The length of the rope outside the barn is $20 m$ which lets the cow to graze a half circle area then when the reaches to the north west of the barn (top left corner) he can graze the north part of the barn but then he has only $10 m$ of rope to turn right which makes a grazing area of quarter of a circle. – Seyed Oct 25 '17 at 10:22

*not to scale

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

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.