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

Question

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

Answer 1

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.