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A Reader's Solution

September 1999

The first correct solution we received for Puzzle No 8 - The Gobbling Goat was this one from V. Shashidhar.

You can see the more detailed PASS Maths solution for a step-by-step explanation of how to tackle the problem.


The answer to the question is 115.8 metres.

Solution diagram 1

It comes as a root to the equation

  \begin{equation}  {2r^2 \arcsin (\frac{L}{2r})} - {\frac{L}{2}\sqrt {4 r^2 - L^2}} + {L^2 \arccos (\frac{L}{2r})} = {\frac{\pi r^2}{2}} \end{equation}   (1)

where $r=100$ metres and $L$ is the length of the rope used to tie the goat.

The derivation is as follows (see figure):

  $\displaystyle \theta  $ $\displaystyle = $ $\displaystyle  2 \arccos \frac{L}{2 r}  $    
  $\displaystyle \phi  $ $\displaystyle = $ $\displaystyle  4 \arcsin \frac{L}{2 r}  $    
  $\displaystyle Area(CDEB)  $ $\displaystyle = $ $\displaystyle  \frac{\theta L^2}{2}  $    
  $\displaystyle Area(DCF)  $ $\displaystyle = $ $\displaystyle  Area(ADFC) - Area(ADC)  $    
  $\displaystyle  $ $\displaystyle = $ $\displaystyle  {r^2 \arcsin \frac{L}{2 r}} - {\frac{L}{4} \sqrt {4 r^2 - L^2}}  $    
  $\displaystyle Area(EDFCGB)  $ $\displaystyle = $ $\displaystyle  Area(CDEB) + 2 Area(DCF)  $    
  $\displaystyle  $ $\displaystyle = $ $\displaystyle  \frac{\pi r^2}{2}  $    

This implies equation (1).

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