Goat

Meadow is a circle with radius r = 19 m. How long must a rope to tie a goat to the pin on the perimeter of the meadow to allow goat eat half of meadow?

Result

x =  22.02 m

Solution:

S1+S2=12S S=πr2 S1=12x2(αsinα) S2=12r2((2π2α)sin(2π2α))  x=2rcos(α/2)   α=tanαπ2cosα    x=1.158728419=22.02 mS_1 + S_2 = \dfrac12 S \ \\ S = \pi r^2 \ \\ S_1 = \dfrac12x^2(\alpha -\sin \alpha) \ \\ S_2 = \dfrac12r^2((2\pi - 2\alpha) -\sin (2\pi - 2\alpha)) \ \\ \ \\ x = 2r \cos (\alpha/2) \ \\ \ \\ \ \\ \alpha = \tan \alpha - \dfrac{ \pi }{ 2 \cos \alpha } \ \\ \ \\ \ \\ \ \\ x = 1.1587284 \cdot 19 = 22.02 \ \text{m}



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