| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Topic | Differential equations |
| Type | Exponential growth/decay - approach to limit (dN/dt = k(N - N₀)) |
| Difficulty | Standard +0.3 This is a standard C4 differential equations question involving rates of change. Part (a) requires setting up the differential equation from a word problem (straightforward bookwork). Part (b) involves separating variables and integrating a linear form, which is routine C4 technique. Part (c) is trivial once part (b) is complete. The question is slightly easier than average because the setup is clear, the algebra is manageable, and it follows a standard template for this topic. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)4.10a General/particular solutions: of differential equations4.10c Integrating factor: first order equations |
Liquid is poured into a container at a constant rate of 30 cm$^3$ s$^{-1}$. At time $t$ seconds liquid is leaking from the container at a rate of $\frac{1}{5}V$ cm$^3$ s$^{-1}$, where $V$ cm$^3$ is the volume of liquid in the container at that time.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$-15 \frac{dV}{dt} = 2V - 450.$$ [3]
\end{enumerate}
Given that $V = 1000$ when $t = 0$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the solution of the differential equation, in the form $V = f(t)$. [7]
\item Find the limiting value of $V$ as $t \to \infty$. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q30 [11]}}