| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2008 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Variable acceleration (1D) |
| Type | Related rates with explicitly given non-geometric algebraic relationships |
| Difficulty | Moderate -0.8 This is a straightforward related rates problem with clear scaffolding. Part (i) is trivial substitution, part (ii) is basic differentiation of k/V, and part (iii) applies the chain rule dP/dt = (dP/dV)(dV/dt) with all values given. Requires only routine C3 differentiation techniques with no problem-solving insight needed. |
| Spec | 1.07b Gradient as rate of change: dy/dx notation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
4 When the gas in a balloon is kept at a constant temperature, the pressure $P$ in atmospheres and the volume $V \mathrm {~m} ^ { 3 }$ are related by the equation
$$P = \frac { k } { V }$$
where $k$ is a constant. [This is known as Boyle's Law.]\\
When the volume is $100 \mathrm {~m} ^ { 3 }$, the pressure is 5 atmospheres, and the volume is increasing at a rate of $10 \mathrm {~m} ^ { 3 }$ per second.\\
(i) Show that $k = 500$.\\
(ii) Find $\frac { \mathrm { d } P } { \mathrm {~d} V }$ in terms of $V$.\\
(iii) Find the rate at which the pressure is decreasing when $V = 100$.
\hfill \mbox{\textit{OCR MEI C3 2008 Q4 [7]}}