| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Tank/container - constant cross-section (cuboid/cylinder) |
| Difficulty | Moderate -0.3 This is a standard C4 differential equations question with straightforward separation of variables and application of boundary conditions. While it requires multiple steps across four parts, each individual step follows routine techniques (relating V to h, separating variables, integrating √h, applying initial conditions). The mathematical manipulations are mechanical rather than requiring insight, making it slightly easier than average for A-level. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)1.08l Interpret differential equation solutions: in context |
Fluid flows out of a cylindrical tank with constant cross section. At time $t$ minutes, $t \geq 0$, the volume of fluid remaining in the tank is $V$ m$^3$. The rate at which the fluid flows, in m$^3$ min$^{-1}$, is proportional to the square root of $V$.
\begin{enumerate}[label=(\alph*)]
\item Show that the depth $h$ metres of fluid in the tank satisfies the differential equation
$$\frac{dh}{dt} = -k\sqrt{h}, \quad \text{where } k \text{ is a positive constant.}$$
[3]
\item Show that the general solution of the differential equation may be written as
$$h = (A - Bt)^2, \quad \text{where } A \text{ and } B \text{ are constants.}$$
[4]
Given that at time $t = 0$ the depth of fluid in the tank is 1 m, and that 5 minutes later the depth of fluid has reduced to 0.5 m,
\item find the time, $T$ minutes, which it takes for the tank to empty.
[3]
\item Find the depth of water in the tank at time $0.5T$ minutes.
[2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q2 [12]}}