So if I work out what that is, it’s going to be 2cos alpha over 2, and then that’s going to be the length of the rope that I need to tether the goat to the side of the field so that he can eat half the field.
I’m going to tell you about the goat problem. It’s not the greatest of all time, but it’s good enough. It’s a school-level problem, but it’s hard to solve. For a long time, we only had an approximation of the answer, until recently when we found an exact answer.
The problem involves a goat tethered to the side of a circular field with a radius of 1. The rope has a length of r. The question is: how long should the rope be if you want the goat to graze on half the field? It feels like something you could solve with school-level maths.
To calculate the area of the field, you use the traditional formula for working out the area of a circle, which is pi. To make the goat eat half the field, the length of the rope needs to be 2cos alpha over 2, where alpha is the angle made. It looks like we have a solution to the problem with the goat eating the grass! We can calculate the angle alpha by setting the area the goat is eating to be half the field, which is $\pi/2$. This results in an approximate angle of 109 degrees point 1885… To work out the length of the rope, we can use the formula $2 \cos \alpha/2$ which gives us an approximate answer of 1.15872847301… which is a little bit longer than the radius of the field. My friends recently found an exact answer to the 3-dimensional version of the Goat Problem, which is surprisingly hard. This answer is a monster expression, defined as a = 16/9 + 4/3 + (4 + sqrt(8)^(1/3)) + (4 - sqrt(8)^(1/3)) and the length of the rope thread for the bird is 2/3 + (1/2)sqrt(a) - (sqrt(16/3) - a) + (128/27)sqrt(a). It was originally approximated by Marshall Fraser, but now an exact answer has been found. Interestingly, in higher dimensions, the answer tends to the square root of 2. It’s like a Sudoku - every row, column, and the diagonals should include a club, a heart, a spade, and a diamond. Try it out, it should be the same kind of thing as a Sudoku - there is a solution I’m not trying to catch you out.