OCR C4 — Question 5 11 marks

Exam BoardOCR
ModuleC4 (Core Mathematics 4)
Marks11
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicDifferential equations
TypeNewton's law of cooling
DifficultyModerate -0.3 This is a standard Newton's law of cooling problem with straightforward setup and solution. Part (i) requires translating the word statement into a differential equation (routine), part (ii) involves separating variables, integrating, and applying two boundary conditions to find constants (standard C4 technique), and part (iii) is a simple substitution. While it requires multiple steps, each follows a well-practiced procedure with no novel insight needed, making it slightly easier than average.
Spec1.08k Separable differential equations: dy/dx = f(x)g(y)1.08l Interpret differential equation solutions: in context
  1. A bath is filled with hot water which is allowed to cool. The temperature of the water is \(\theta ^ { \circ } \mathrm { C }\) after cooling for \(t\) minutes and the temperature of the room is assumed to remain constant at \(20 ^ { \circ } \mathrm { C }\).
Given that the rate at which the temperature of the water decreases is proportional to the difference in temperature between the water and the room,
  1. write down a differential equation connecting \(\theta\) and \(t\). Given also that the temperature of the water is initially \(37 ^ { \circ } \mathrm { C }\) and that it is \(36 ^ { \circ } \mathrm { C }\) after cooling for four minutes,
  2. find, to 3 significant figures, the temperature of the water after ten minutes. Advice suggests that the temperature of the water should be allowed to cool to \(33 ^ { \circ } \mathrm { C }\) before a child gets in.
  3. Find, to the nearest second, how long a child should wait before getting into the bath.
\begin{enumerate}
  \item A bath is filled with hot water which is allowed to cool. The temperature of the water is $\theta ^ { \circ } \mathrm { C }$ after cooling for $t$ minutes and the temperature of the room is assumed to remain constant at $20 ^ { \circ } \mathrm { C }$.
\end{enumerate}

Given that the rate at which the temperature of the water decreases is proportional to the difference in temperature between the water and the room,\\
(i) write down a differential equation connecting $\theta$ and $t$.

Given also that the temperature of the water is initially $37 ^ { \circ } \mathrm { C }$ and that it is $36 ^ { \circ } \mathrm { C }$ after cooling for four minutes,\\
(ii) find, to 3 significant figures, the temperature of the water after ten minutes.

Advice suggests that the temperature of the water should be allowed to cool to $33 ^ { \circ } \mathrm { C }$ before a child gets in.\\
(iii) Find, to the nearest second, how long a child should wait before getting into the bath.\\

\hfill \mbox{\textit{OCR C4  Q5 [11]}}
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