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, α=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)≈36. Given the range of possible solutions to this problem there is exactly one other option: V(2.511167869)≈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 ∑x correct to 9 decimal places.
浪费了平方数空间的谷仓
农夫弗雷德在他的农场上造了一个新谷仓，当他发现这个谷仓竟然是圆柱形时，这个有着对正方形狂热的喜好的农夫非常愤怒。谷仓制造商的代表昆廷解释道，他们公司从来都只制造圆柱形的谷仓，同时他指出，这个谷仓的地基是正方形的。弗雷德并没有因此而释然，仍然坚持要拆掉这个谷仓。
机灵鬼昆廷脑子一转，解释道，当谷物从谷仓的上面往下倒时，顶部会形成一个锥形，锥顶与水平面的角度称为静止角。例如，当静止角α=30，且锥顶位于谷仓的中央时，顶部将是一个标准的圆锥形，而谷仓的直径是6米，所以谷仓中浪费的空间是约32.648388556 m3。然而，如果锥顶的位置距离谷仓的中央为x米时，将会形成一个奇特的弯曲斜面。他给弗雷德画了一张示意图。
我们用V(x)来表示浪费的空间，单位是立方米。如果x=1.114785284，保留到3的平方位小数，则浪费的空间V(1.114785284)≈36。在可行的范围内，恰好还有另外一个选择V(2.511167869)≈49。这就好比平方数，也就是正方形数，是谷仓之王，高高地站在你的谷物的顶上。
弗雷德对于这个优雅的解释眼前一亮，但是当他更仔细地观察昆廷的示意图和计算后，他的笑容马上无影无踪了。弗雷德向昆廷指出，谷物的半径而非直径是6米，而且谷物所形成的静止角是40度。然而，如果昆廷仍然能够找到合适的解，弗雷德将会愿意保留这个谷仓。
帮助机灵鬼昆廷满足弗雷德折腾人的正方形狂热，找出所有使得浪费的空间为平方数的x值，并计算出∑x，保留9位小数。