Chruser
2020-12-13, 07:45 AM
https://i.imgur.com/X1EVwOu.png
Question: How big must r be chosen in the diagram, in order for the red area to equal one half of the area of the circle?
The above problem (https://en.m.wikipedia.org/wiki/Goat_problem) was first posed in 1748, and it has been known since then that r is approximately equal to 1.1578. Apparently it just took us 272 years to figure out an exact solution (https://www.quantamagazine.org/mathematician-solves-centuries-old-grazing-goat-problem-exactly-20201209/) to it. Namely,
r=2\,\cos\,\left( \frac{1}{2}\cdot\frac{ \oint_{|z-3 \pi / 8|=\pi / 4} z /(\sin z-z \cos z-\pi / 2) \, dz}{\oint_{|z-3 \pi / 8|=\pi / 4} 1 /(\sin z-z \cos z-\pi / 2)\, dz}\right)
where the integrals are contour integrals (https://en.m.wikipedia.org/wiki/Contour_integration).
Good job, humanity.
In all seriousness, it's fascinating how difficult trivial-looking math problems can become once you start to experiment a little with the functions/constants/boundary conditions/shapes/etc. Another good example of this is the calculation of the arc length of a curve.
Specifically, if y = f(x) is a continuously differentiable function on the interval (a,b), then its arc length s between a and b is
s=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}dx.
There are some simple exact solutions to the above integral, but let's just say that there are reasons why most calculus textbooks stick to examples where f(x) describes either a line, parabola, circle or hyperbolic cosine curve...
https://en.wikipedia.org/wiki/Arc_length
In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary.
Question: How big must r be chosen in the diagram, in order for the red area to equal one half of the area of the circle?
The above problem (https://en.m.wikipedia.org/wiki/Goat_problem) was first posed in 1748, and it has been known since then that r is approximately equal to 1.1578. Apparently it just took us 272 years to figure out an exact solution (https://www.quantamagazine.org/mathematician-solves-centuries-old-grazing-goat-problem-exactly-20201209/) to it. Namely,
r=2\,\cos\,\left( \frac{1}{2}\cdot\frac{ \oint_{|z-3 \pi / 8|=\pi / 4} z /(\sin z-z \cos z-\pi / 2) \, dz}{\oint_{|z-3 \pi / 8|=\pi / 4} 1 /(\sin z-z \cos z-\pi / 2)\, dz}\right)
where the integrals are contour integrals (https://en.m.wikipedia.org/wiki/Contour_integration).
Good job, humanity.
In all seriousness, it's fascinating how difficult trivial-looking math problems can become once you start to experiment a little with the functions/constants/boundary conditions/shapes/etc. Another good example of this is the calculation of the arc length of a curve.
Specifically, if y = f(x) is a continuously differentiable function on the interval (a,b), then its arc length s between a and b is
s=\int_a^b \sqrt{1+\left(\frac{dy}{dx}\right)^2}dx.
There are some simple exact solutions to the above integral, but let's just say that there are reasons why most calculus textbooks stick to examples where f(x) describes either a line, parabola, circle or hyperbolic cosine curve...
https://en.wikipedia.org/wiki/Arc_length
In most cases, including even simple curves, there are no closed-form solutions for arc length and numerical integration is necessary.