A goat

In the square garden of side (a), a goat is tied in the middle of one side. Calculate the length of the rope (r) so that the goat grazes exactly half the garden. If r = c * a, find the constant c.

Final Answer:

c =  1.41421356

Step-by-step explanation:

$$\begin{gathered} S=a^2/2 \\ S=S_1+S_2 \\ {S}_1=a\cdot b=a^2/2\tan (x/2) \\ {S}_2=r^2/2(x-\sin x) \\ r=\sqrt{b^2+(a/2{)}^2} \\ r=c\cdot a \\ a=1 \\ S=a^2/2=1^2/2=\frac{1}{2}=0.5 \\ 0.5=b+(b^2+0.5^2)/2\cdot (x-\sin x) \\ b=0.5/\tan (x/2) \\ 0.5=0.5/\tan (x/2)+((0.5/\tan (x/2){)}^2+0.5^2)/2\cdot (x-\sin x) \\ c=\sqrt{2}=1.41421356 \end{gathered}$$

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