Goat

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

Correct answer:

x =  22.02 m

Step-by-step explanation:

$$\begin{gathered} S_1+S_2=\frac{1}{2}S \\ S=\pi r^2 \\ {S}_1=\frac{1}{2}{x}^2(\alpha -\sin \alpha ) \\ {S}_2=\frac{1}{2}{r}^2((2\pi -2\alpha )-\sin (2\pi -2\alpha )) \\ x=2r\cos (\alpha /2) \\ \alpha =\tan \alpha -\frac{\pi }{2\cos \alpha } \\ x=1.1587284\cdot 19=22.02\text{m} \end{gathered}$$

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