Re: Math Challenge I could not solve Message #10 Posted by PGILLET on 7 Jan 2013, 2:28 a.m., in response to message #7 by Thomas Klemm
That's how I saw the problem:
In the lensshaped piece you can see an isosceles triangle with sides (1,1,r) and angles (a,a,pi2*a).
Divided in two rightangled triangles, it leads to 1*cos(pi2*a)=d and r*cos(a)=1d => d=1r*r/2.
The grazed surface can then be expressed as
S=2*(integ(sqrt(1x*x),x,1r*r/2,1)+integ(sqrt(r*rx*x),x,r*r/2,r)).
But we know that integ(sqrt(r*rx*x),x)=x/2*sqrt(r*rx*x)+r*r/2*arcsin(x/r).
So developing and simplifying S naturally leads to
S=arccos(1r*r/2)+r*r*arccos(r/2)r/2*sqrt(4r*r).
(It's just another way to the same formula as in the http://mathworld.wolfram.com/GoatProblem.html web site).
Solving S=pi/2 on r also :) leads to r=1.15872847...
