Question 86
A wealthy king has his blacksmith fashion him a large cup, whose inside is a cone of height $9$ inches and base diameter $6$ inches (that is, the opening at the top of the cup is $6$ inches in diameter). At one of his many feasts, he orders the mug to be filled to the brim with cranberry juice.
For each positive integer $n$, the king stirs his drink vigorously and takes a sip such that the height of fluid left in his cup after the sip goes down by $\frac{1}{n^{2}}$ inches. Shortly afterwards, while the king is distracted, the court jester adds pure Soylent to the cup until it's once again full. The king takes sips precisely every minute, and his first sip is exactly one minute after the feast begins.
As time progresses, the amount of juice consumed by the king (in cubic inches) approaches a number $r$. Find $r$.
For each positive integer $n$, the king stirs his drink vigorously and takes a sip such that the height of fluid left in his cup after the sip goes down by $\frac{1}{n^{2}}$ inches. Shortly afterwards, while the king is distracted, the court jester adds pure Soylent to the cup until it's once again full. The king takes sips precisely every minute, and his first sip is exactly one minute after the feast begins.
As time progresses, the amount of juice consumed by the king (in cubic inches) approaches a number $r$. Find $r$.