N2025 P2 Q4

The diagram shows a hemispherical bowl, of inside radius \(15\text{ cm}\), fixed so its circular rim is horizontal. When the depth of water in the bowl is \(h\text{ cm}\), the volume, \(V\text{ cm}^3\), of water in the bowl is given by \(V=\dfrac{1}{3}\pi h^2(45-h).\)

The bowl is initially empty, and water is poured into the bowl at the constant rate of \(10\pi\text{ cm}^3\) per second.

1. Find the time taken to fill the bowl.[2]
2. Find the exact rate at which the depth of water is increasing when the depth of water is \(12\text{ cm}\).[4]

The bowl is emptied, and water is poured into the bowl at the variable rate of \(10\pi t\text{ cm}^3\) per second, where \(t\) seconds is the time from when pouring started.

3. Find the time taken to pour \(972\pi\text{ cm}^3\) of water into the bowl.

   [3]

4. Find the rate at which the depth of water is increasing when the volume of water in the bowl is \(972\pi\text{ cm}^3\).

   [4]