Suppose there is a cow that is tied to the corner of a barn with a rope as shown in the

figure below. One side of square barn measures 50 yards
and all the doors on the barn are closed, so the
cow cannot enter the barn (he must walk around the sides only).
How
large of an area
does the co
w have
to walk around and eat
if the length of rope is
1 40 yards
2 75 yards
3 125 yards

First one is easy.

The cow can cover 3/4 of a circle with radius 40
(the missing quarter is the barn)
area = (3/4)(π(40^2)
= 1200π yds^2

2nd one:
let's tie the cow up at the top left corner.
with a rope of 75 yds, the cow can cover (3/4)(π(75^2)
or 4218.75π yds^2
Draw an extension of the barn at the top right.
The cow will be able to go 75 yds along that line and then around the corner she can reach the vertical side of the barn 25 yds down that side.
In other words she can cover 1/4 of a smaller circle with radius 25
this will give her an additional (1/4)(π(25^2))
or 156.25π yds^2
But the same will happen at the bottom right corner of the barn
so total area she can cover = 4218.75π + 156.25π +156.25π
= 4531.25π yds^2

3rd one:
I suggest you actually draw in the circles she can cover.
Notice with a rope of 125 yds, the cow can actually come around 2 sides of the barn in either directions and then 25 yds along the third side,
but the radius will change at the corners.

To find out how large of an area the cow has to walk around and eat, we need to calculate the perimeter of the square barn and add the length of the rope to it.

Given dimensions:
- Side length of the square barn = 50 yards
- Length of the rope = 140 yards, 75 yards, or 125 yards (three scenarios)

1. For the scenario where the length of the rope is 140 yards:
- The perimeter of the square barn = 4 * side length = 4 * 50 = 200 yards
- Add the length of the rope: 200 + 140 = 340 yards
- The cow has to walk around and eat an area of 340 square yards.

2. For the scenario where the length of the rope is 75 yards:
- The perimeter of the square barn = 4 * side length = 4 * 50 = 200 yards
- Add the length of the rope: 200 + 75 = 275 yards
- The cow has to walk around and eat an area of 275 square yards.

3. For the scenario where the length of the rope is 125 yards:
- The perimeter of the square barn = 4 * side length = 4 * 50 = 200 yards
- Add the length of the rope: 200 + 125 = 325 yards
- The cow has to walk around and eat an area of 325 square yards.

Therefore, the cow has to walk around and eat an area of 340 square yards when the length of the rope is 140 yards, 275 square yards when the length of the rope is 75 yards, and 325 square yards when the length of the rope is 125 yards.