| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Applied differentiation |
| Type | Spreading stain or growing patch area |
| Difficulty | Moderate -0.8 This is a straightforward applied differentiation question requiring substitution to find constants, basic chain rule differentiation of a quadratic expression, and evaluation at a point. All steps are routine C1 techniques with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part structure. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
Some ink is poured onto a piece of cloth forming a stain that then spreads.
The area of the stain, $A$ cm$^2$, after $t$ seconds is given by
$$A = (p + qt)^2,$$
where $p$ and $q$ are positive constants.
Given that when $t = 0$, $A = 4$ and that when $t = 5$, $A = 9$,
\begin{enumerate}[label=(\roman*)]
\item find the value of $p$ and show that $q = \frac{1}{5}$, [5]
\item find $\frac{\mathrm{d}A}{\mathrm{d}t}$ in terms of $t$, [3]
\item find the rate at which the area of the stain is increasing when $t = 15$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR C1 Q6 [10]}}