| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Newton's law of cooling |
| Difficulty | Standard +0.3 This is a standard C4 differential equations question with clear scaffolding through multiple parts. Part (b) requires separating variables and integrating (routine C4 technique), parts (c) and (d) involve substituting boundary conditions and solving logarithmic equations (standard applications), and part (e) is a simple comparison. The question guides students through each step with no novel insight required, making it slightly easier than average for C4. |
| Spec | 1.06i Exponential growth/decay: in modelling context1.08k Separable differential equations: dy/dx = f(x)g(y) |
Scientists can estimate the time elapsed since an animal died by measuring its body temperature.
\begin{enumerate}[label=(\alph*)]
\item Assuming the temperature goes down at a constant rate of 1.5 degrees Fahrenheit per hour, estimate how long it will take for the temperature to drop
\begin{enumerate}[label=(\Alph*)]
\item from 98°F to 89°F,
\item from 98°F to 80°F. [2]
\end{enumerate}
\end{enumerate}
In practice, rate of temperature loss is not likely to be constant. A better model is provided by Newton's law of cooling, which states that the temperature $\theta$ in degrees Fahrenheit $t$ hours after death is given by the differential equation
$$\frac{d\theta}{dt} = -k(\theta - \theta_0),$$
where $\theta_0$°F is the air temperature and $k$ is a constant.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show by integration that the solution of this equation is $\theta = \theta_0 + Ae^{-kt}$, where $A$ is a constant. [5]
\end{enumerate}
The value of $\theta_0$ is 50, and the initial value of $\theta$ is 98. The initial rate of temperature loss is 1.5°F per hour.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find $A$, and show that $k = 0.03125$. [4]
\item Use this model to calculate how long it will take for the temperature to drop
\begin{enumerate}[label=(\Alph*)]
\item from 98°F to 89°F,
\item from 98°F to 80°F. [5]
\end{enumerate}
\item Comment on the results obtained in parts (i) and (iv). [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C4 Q2 [17]}}