| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2020 |
| Session | October |
| Marks | 6 |
| Topic | Connected Rates of Change |
| Type | Pile or heap: height rate from volume rate |
| Difficulty | Standard +0.3 This is a straightforward related rates problem requiring chain rule differentiation and substitution. Part (i) is routine calculus (differentiating a composite function), and part (ii) applies the standard related rates formula dV/dt = (dV/dh)(dh/dt). The algebra is slightly involved but follows a standard template with no novel insight required. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
Earth is being added to a pile so that, when the height of the pile is $h$ metres, its volume is $V$ cubic metres, where
$$V = (h^6 + 16)^2 - 4.$$
\begin{enumerate}[label=(\roman*)]
\item Find the value of $\frac{dV}{dh}$ when $h = 2$.
[3]
\item The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when $h = 2$. Give your answer correct to 2 significant figures.
[3]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2020 Q9 [6]}}