Answer to Question #178124 in Calculus for Kopano

Question #178124

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.


1
Expert's answer
2021-04-15T07:40:15-0400


Find the area of the field:


S1=100100=10000m2S_1=100\cdot100=10000 m^2

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


S2=14πr2S_2 =\frac{1}{4}\pi r^2

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


14πr2=5000\frac{1}{4}\pi r^2=5000πr2=45000\pi r^2 =4\cdot 5000πr2=20000\pi r^2 =20000r2=20000πr^2 =\frac{20000}{\pi}r=20000π=1002π79.8mr=\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 r79.8mr\approx79.8 m



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