Filling up spheres

Calculus Level 4

A solid spherical tank of inner radius 3 m (thickness irrelevant) is being filled with water at a constant rate of \( \pi \: \:m^{3}/\text{min} \) through a hole at its apex.

Determine how fast the level of the water is increasing (in \(m\: /min \)) at the instant the tank accumulates a volume of \(\large \frac{325}{24}\pi \: \text{m}^{3} \) of water. Assume that the water level at any point on the surface is always equal, that is, no waves occur.

The rate can be expressed in the form \( \frac {A}{B} \) where \(A\) and \(B\) are coprime integers. Input your answer as \( A + B \).

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