| Exam Board | Edexcel |
|---|---|
| Module | P4 (Pure Mathematics 4) |
| Year | 2022 |
| Session | October |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Spherical geometry differential equations |
| Difficulty | Standard +0.3 This is a standard differential equations modelling question requiring separation of variables and using initial/boundary conditions. The setup is clearly stated, the integration is straightforward (∫r² dr), and finding constants from given conditions follows a routine procedure. Slightly easier than average due to clear problem structure and standard technique application. |
| Spec | 1.07t Construct differential equations: in context4.10c Integrating factor: first order equations |
A spherical ball of ice of radius 12 cm is placed in a bucket of water.
In a model of the situation,
• the ball remains spherical as it melts
• $t$ minutes after the ball of ice is placed in the bucket, its radius is $r$ cm
• the rate of decrease of the radius of the ball of ice is inversely proportional to the square of the radius
• the radius of the ball of ice is 6 cm after 15 minutes
Using the model and the information given,
\begin{enumerate}[label=(\alph*)]
\item find an equation linking $r$ and $t$, [5]
\item find the time taken for the ball of ice to melt completely, [2]
\item On Diagram 1 on page 27, sketch a graph of $r$ against $t$. [1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel P4 2022 Q10 [8]}}