| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Tank/container - constant cross-section (cuboid/cylinder) |
| Difficulty | Standard +0.3 This is a standard separable differential equations question with straightforward setup and solution. Part (a) requires relating volume rate to height rate using V=πr²h (basic differentiation), while part (b) involves routine separation of variables and integration of h^(-1/2). The context is familiar and all steps are textbook-standard, making it slightly easier than average. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
\includegraphics{figure_3}
Figure 3 shows a large vertical cylindrical tank containing a liquid. The radius of the circular cross-section of the tank is 40 cm. At time $t$ minutes, the depth of liquid in the tank is $h$ centimetres. The liquid leaks from a hole $P$ at the bottom of the tank.
The liquid leaks from the tank at a rate of $32\pi \sqrt{h}$ cm$^3$ min$^{-1}$.
\begin{enumerate}[label=(\alph*)]
\item Show that at time $t$ minutes, the height $h$ cm of liquid in the tank satisfies the differential equation
$$\frac{dh}{dt} = -0.02\sqrt{h}$$
[4]
\item Find the time taken, to the nearest minute, for the depth of liquid in the tank to decrease from 100 cm to 50 cm.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2013 Q8 [9]}}