| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | First-order integration |
| Difficulty | Moderate -0.3 This is a straightforward C4 differential equations question requiring direct integration of an exponential function, substitution of initial conditions, and solving for constants. The integration is standard (∫e^(-0.2t) dt), and all parts follow mechanically from the previous one with no conceptual challenges or novel problem-solving required. Slightly easier than average due to the explicit form of dy/dt and clear step-by-step structure. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(\int dy = \int -ke^{-0.2t} dt\) | M1 | |
| \(y = 5ke^{-0.2t} + c\) | A1 | |
| \(t = 0, y = 2 \Rightarrow 2 = 5k + c, \quad c = 2 - 5k\) | M1 | |
| \(\therefore y = 5ke^{-0.2t} - 5k + 2\) | A1 | |
| (b) \(t = 2, y = 1.6 \Rightarrow 1.6 = 5ke^{-0.4} - 5k + 2\) | M1 | |
| \(k = \frac{-0.4}{5e^{-0.4} - 5} = 0.2427\) (4sf) | M1 A1 | |
| (c) as \(t \to \infty, y \to h\) (in metres) | M1 | |
| \(\therefore "h" = -5k + 2 = 0.787\) m \(= 78.7\) cm ∴ \(h = 79\) | M1 A1 | (10) |
**(a)** $\int dy = \int -ke^{-0.2t} dt$ | M1 |
$y = 5ke^{-0.2t} + c$ | A1 |
$t = 0, y = 2 \Rightarrow 2 = 5k + c, \quad c = 2 - 5k$ | M1 |
$\therefore y = 5ke^{-0.2t} - 5k + 2$ | A1 |
**(b)** $t = 2, y = 1.6 \Rightarrow 1.6 = 5ke^{-0.4} - 5k + 2$ | M1 |
$k = \frac{-0.4}{5e^{-0.4} - 5} = 0.2427$ (4sf) | M1 A1 |
**(c)** as $t \to \infty, y \to h$ (in metres) | M1 |
$\therefore "h" = -5k + 2 = 0.787$ m $= 78.7$ cm ∴ $h = 79$ | M1 A1 | (10)5. At time $t = 0$, a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, $y$ metres, after $t$ hours satisfies the differential equation
$$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k \mathrm { e } ^ { - 0.2 t }$$
where $k$ is a positive constant,
\begin{enumerate}[label=(\alph*)]
\item Find an expression for $y$ in terms of $k$ and $t$.
Given that two hours after being filled the depth of water in the tank is 1.6 metres,
\item find the value of $k$ to 4 significant figures.
Given also that the hole in the tank is $h \mathrm {~cm}$ above the base of the tank,
\item show that $h = 79$ to 2 significant figures.\\
5. continued
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q5 [10]}}