| Exam Board | AQA |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Topic | Differential equations |
| Type | Spherical geometry differential equations |
| Difficulty | Moderate -0.3 This is a straightforward differential equations question requiring standard techniques: setting up dA/dt = -k, integrating to find A(t), using initial conditions to find constants, and interpreting the result. The sphere surface area formula is given, and part (b)(i) guides students through the calculation. While it requires multiple steps, each step follows routine procedures with no novel insight needed, making it slightly easier than average. |
| Spec | 1.07t Construct differential equations: in context1.08l Interpret differential equation solutions: in context |
7 A giant snowball is melting. The snowball can be modelled as a sphere whose surface area is decreasing at a constant rate with respect to time. The surface area of the sphere is $A \mathrm {~cm} ^ { 2 }$ at time $t$ days after it begins to melt.
\begin{enumerate}[label=(\alph*)]
\item Write down a differential equation in terms of the variables $A$ and $t$ and a constant $k$, where $k > 0$, to model the melting snowball.
\item \begin{enumerate}[label=(\roman*)]
\item Initially, the radius of the snowball is 60 cm , and 9 days later, the radius has halved. Show that $A = 1200 \pi ( 12 - t )$.\\
(You may assume that the surface area of a sphere is given by $A = 4 \pi r ^ { 2 }$, where $r$ is the radius.)
\item Use this model to find the number of days that it takes the snowball to melt completely.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C4 2011 Q7 [7]}}