A-LevelMathematicsDifferential equations9709/31 · Oct/Nov 2024 · Q1010 Marks
A large cylindrical tank is used to store water. The base of the tank is a circle of radius 4 metres. At time t minutes, the depth of the water in the tank is h metres. There is a tap at the bottom of the tank. When the tap is open, water flows out of the tank at a rate proportional to the square root of the volume of water in the tank. [Figure 10.1]
- (a)Show that dh/dt = -λ√h, where λ is a positive constant.[4]
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Mark scheme
Marking points from the official mark scheme (Oct/Nov 2024). The mark or mark type is shown in brackets.
- [B1] dV/dt = ±k√V or dV/dt = 16π dh/dt SOI
- [M1] Correct use of chain rule and V=16πh OE, e.g. dV/dt = 16π dh/dt
- [A1] Obtain dh/dt = (-k√16πh)/(16π) Any equivalent form in terms of h.
- [A1] Obtain dh/dt = -λ√h where λ = k/(4√π) is constant Obtain given answer from full and correct working.
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