21M.1.AHL.TZ1.12

pestleMathematicsAIHLPaper 121M· ahl-5-14-setting-up-a-de-solve-by-separating-variablessource ↗

A tank of water initially contains $400$ litres. Water is leaking from the tank such that after $10$ minutes there are $324$ litres remaining in the tank.

The volume of water, $V$ litres, remaining in the tank after $t$ minutes, can be modelled by the differential equation

$\frac{d V}{d t} = - k \sqrt{V}$ , where $k$ is a constant.

Show that $V = {(20 - \frac{t}{5})}^{2}$ .

[6]

Find the time taken for the tank to empty.

[2]

Markscheme / solution

$\frac{d V}{d t} = - k V^{\frac{1}{2}}$

use of separation of variables (M1)

$\Rightarrow \int V^{- \frac{1}{2}} d V = \int - k d t$ A1

$2 V^{\frac{1}{2}} = - k t (+ c)$ A1

considering initial conditions $40 = c$ A1

$2 \sqrt{324} = - 10 k + 40$

$\Rightarrow k = 0.4$ A1

$2 \sqrt{V} = - 0.4 t + 40$

$\Rightarrow \sqrt{V} = 20 - 0.2 t$ A1

Note: Award A1 for any correct intermediate step that leads to the AG.

$\Rightarrow V = {(20 - \frac{t}{5})}^{2}$ AG

Note: Do not award the final A1 if the AG line is not stated.

[6 marks]

$0 = {(20 - \frac{t}{5})}^{2} \Rightarrow t = 100$ minutes (M1)A1

[2 marks]

Examiners’ report

[N/A]