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Draining tank that never empties

Draining tank that never empties

IP 4
9 marks

A tank holds \(500\) litres of water. At time \(t=0\) a valve is opened, and the volume \(V\) litres after \(t\) minutes is modelled by \(\dfrac{\mathrm dV}{\mathrm dt}=-20e^{-0.1t}\).

  1. Find \(V\) in terms of \(t\).[3]
  2. Show that, according to the model, the tank never empties, and state the volume it approaches.[2]
  3. A gauge reads \(385\) litres after \(10\) minutes. Compare this with the model and suggest one reason for the difference.[2]
  4. Find the time at which half of the water that will eventually drain has drained.[2]

Solution:

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Answer:(a) \(V=300+200e^{-0.1t}\) (b) \(V>300\) for all \(t\); approaches \(300\) litres (c) model gives about \(374\) litres, the gauge \(385\); water drains more slowly than modelled (d) \(t=10\ln2\approx6.93\) minutes

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