Answer to Question #178124 in Calculus for Kopano

Question

A goat is tied by a rope of length r metres to the corner of a square field with sides of length 100 m,. Find r if the goat is able to reach exactly half the grass in the field.

Answer 1

Find the area of the field:

$$S_1=100\cdot100=10000 m^2$$

A goat is able to reach the area of a quarter of the circle:

$$S_2 =\frac{1}{4}\pi r^2$$

This area is equal to half the area of the field:

$$\frac{1}{4}\pi r^2=5000$$ $$\pi r^2 =4\cdot 5000$$ $$\pi r^2 =20000$$ $$r^2 =\frac{20000}{\pi}$$ $$r=\sqrt{\frac{20000}{\pi}}=100\sqrt{\frac{2}{\pi}}\approx79.8 m$$

Answer: The goat is able to reach exactly half the grass in the field if $r\approx79.8 m$