Question

Four cows are tethered at the four corners of a square field of side 50 m such that each can graze the maximum unshared area. What area will be left ungrazed ?

[ Take $\pi =3.14$]

Answer 1

If they are tethered at the corners they can map out quarter circles within the square. For these to be maximised without sharing the quarter circles must have a radius of $\frac{1}{2}$ of one side of the square, 25m.

Now we have four quarter circles about each corner, touching at the middle of each side.

From here we know that the full area of the field is $50\times 50=2500m^2$

We can calculate the grazed area as 1 circle of radius 25m $(4\times \frac{1}{4}circles)$

The area of this is equal to $\pi \times r^2=3.14\times 25\times 25=1963.50m^2$

Now the ungrazed area is the total minus the ungrazed $=2500-1963=537m^2$