# Problem 431: Square Space Silo
Fred the farmer arranges to have a new storage silo installed on his
farm and having an obsession for all things square he is absolutely
devastated when he discovers that it is circular. Quentin, the
representative from the company that installed the silo, explains that
they only manufacture cylindrical silos, but he points out that it is
resting on a square base. Fred is not amused and insists that it is
removed from his property. Quick thinking Quentin explains that when
granular materials are delivered from above a conical slope is formed
and the natural angle made with the horizontal is called the angle of
repose. For example if the angle of repose, \$\\alpha = 30\$ degrees,
and grain is delivered at the centre of the silo then a perfect cone
will form towards the top of the cylinder. In the case of this silo,
which has a diameter of 6m, the amount of space wasted would be
approximately 32.648388556 m3. However, if grain is delivered at a point
on the top which has a horizontal distance of \$x\$ metres from the
centre then a cone with a strangely curved and sloping base is formed.
He shows Fred a picture. We shall let the amount of space wasted in
cubic metres be given by \$V(x)\$. If \$x = 1.114785284\$, which happens
to have three squared decimal places, then the amount of space wasted,
\$V(1.114785284) \\approx 36\$. Given the range of possible solutions to
this problem there is exactly one other option: \$V(2.511167869)
\\approx 49\$. It would be like knowing that the square is king of the
silo, sitting in splendid glory on top of your grain. Fred's eyes light
up with delight at this elegant resolution, but on closer inspection of
Quentin's drawings and calculations his happiness turns to despondency
once more. Fred points out to Quentin that it's the radius of the silo
that is 6 metres, not the diameter, and the angle of repose for his
grain is 40 degrees. However, if Quentin can find a set of solutions for
this particular silo then he will be more than happy to keep it. If
Quick thinking Quentin is to satisfy frustratingly fussy Fred the
farmer's appetite for all things square then determine the values of
\$x\$ for all possible square space wastage options and calculate
\$\\sum x\$ correct to 9 decimal places.