CAIE A-Level Mathematics 3.8.1 Forming Differential Equations

CAIE A-Level Mathematics 3.8.1 Forming Differential Equations
Cambridge International AS & A Level Mathematics 9709 syllabus for exams in 2028, 2029 and 20302028–2030

Practise converting proportionality and inflow–outflow statements into differential equations and using volume–dimension relationships with the chain rule to obtain the required…

How this is tested

  • introduce a positive proportionality constant with the sign fixed by growth or decay
  • write the net physical rate as inflow minus outflow in the stated variables
  • use dV/dt = (dV/dx)(dx/dt) and given data to determine the constant and final model

Question 10(a)

[Maximum number: 4]

The diagram shows a tank for holding water. The tank is in the shape of a cube of side 50 cm. At time t seconds, the depth of water in the tank is h cmh \mathrm{~cm}. Water is poured into the tank at a rate of 5000 cm3 s−15000 \mathrm{~cm}^{3} \mathrm{~s}^{-1}. Water pours out of the tank through a hole in the bottom at a rate proportional to h2h^{2}.

When h=20, the depth of the water is increasing at a rate of 0.4cms−10.4 \mathrm{cms}^{-1}.

Show that dh dt=500−h2250\frac{\mathrm{d} h}{\mathrm{~d} t}=\frac{500-h^{2}}{250}.