| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Connected Rates of Change |
| Type | Pile or heap: height rate from volume rate |
| Difficulty | Standard +0.3 This is a straightforward connected rates of change question requiring chain rule application. Part (i) is routine differentiation of a composite function, and part (ii) applies the standard formula dh/dt = (dh/dV)(dV/dt). The algebra is slightly more involved than the most basic examples due to the power 1/2 and the h^6 term, but the method is completely standard with no problem-solving insight required. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
4 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
$$V = \left( h ^ { 6 } + 16 \right) ^ { \frac { 1 } { 2 } } - 4$$
(i) Find the value of $\frac { \mathrm { d } V } { \mathrm {~d} h }$ when $h = 2$.\\
(ii) 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.
\hfill \mbox{\textit{OCR C3 2008 Q4 [6]}}