| Exam Board | Edexcel |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 11 |
| 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 related rates problem requiring substitution to find constants, basic differentiation using the chain rule, 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 | 1.02e Complete the square: quadratic polynomials and turning points1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(t = 0, A = 4 \Rightarrow 4 = p^2\) | M1 | |
| \(p > 0 \therefore p = 2\) | A1 | |
| \(t = 5, A = 9 \Rightarrow 9 = (2 + 5q)^2\) | M1 | |
| \(2 + 5q = \pm 3\) | ||
| \(q = \frac{1}{5}(-2 \pm 3)\) | M1 | |
| \(q > 0 \therefore q = \frac{1}{5}\) | A1 | |
| (b) \(A = (2 + \frac{1}{5}t)^2 = 4 + \frac{4}{5}t + \frac{1}{25}t^2\) | M1 A1 | |
| \(\frac{dA}{dt} = \frac{4}{5} + \frac{2}{25}t\) | M1 A1 | |
| (c) \(t = 15 \therefore \frac{dA}{dt} = \frac{4}{5} + \frac{2}{25}(15) = 2 \text{ cm}^2\text{s}^{-1}\) | M1 A1 | (11) |
(a) $t = 0, A = 4 \Rightarrow 4 = p^2$ | M1 |
$p > 0 \therefore p = 2$ | A1 |
$t = 5, A = 9 \Rightarrow 9 = (2 + 5q)^2$ | M1 |
$2 + 5q = \pm 3$ | |
$q = \frac{1}{5}(-2 \pm 3)$ | M1 |
$q > 0 \therefore q = \frac{1}{5}$ | A1 |
(b) $A = (2 + \frac{1}{5}t)^2 = 4 + \frac{4}{5}t + \frac{1}{25}t^2$ | M1 A1 |
$\frac{dA}{dt} = \frac{4}{5} + \frac{2}{25}t$ | M1 A1 |
(c) $t = 15 \therefore \frac{dA}{dt} = \frac{4}{5} + \frac{2}{25}(15) = 2 \text{ cm}^2\text{s}^{-1}$ | M1 A1 | (11)8. Some ink is poured onto a piece of cloth forming a stain that then spreads.
The area of the stain, $A \mathrm {~cm} ^ { 2 }$, after $t$ seconds is given by
$$A = ( p + q t ) ^ { 2 } ,$$
where $p$ and $q$ are positive constants.\\
Given that when $t = 0 , A = 4$ and that when $t = 5 , A = 9$,
\begin{enumerate}[label=(\alph*)]
\item find the value of $p$ and show that $q = \frac { 1 } { 5 }$,
\item find $\frac { \mathrm { d } A } { \mathrm {~d} t }$ in terms of $t$,
\item find the rate at which the area of the stain is increasing when $t = 15$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C1 Q8 [11]}}