Question

There are two sets of parallel lines, their equations being $x\cos \alpha +y\sin \alpha =p$ and $y\cos \alpha -x\cos \alpha =p;p=1,2,3,....,n$ and $\alpha$ is a constant angle. If the number of rectangles formed by these two sets of lines is $225$ then find the value of $n$.

Answer 1

$\frac{n(n+1)(2n+1)}{6}=225\times 6$

$n(2n^2+n+2n+1)=1350$ $[\sum n^2=\frac{n(n+1)(2n+1)}{6}]$

$2n^3+3n^2+n=1350$ $[\therefore 225\times 6=1350]$

By cubic methods,

$n\approx 8$