| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2005 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Optimise geometric shape surface area/volume |
| Difficulty | Standard +0.3 This is a straightforward optimization problem requiring differentiation of a given polynomial expression and solving dV/dr = 0. Part (i) is algebraic manipulation (already guided), and part (ii) is a standard textbook exercise in finding stationary points—slightly easier than average since the expression is already simplified and the differentiation is routine. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{6}{12} = \frac{r}{12-h} \rightarrow h = 12 - 2r\) | M1 A1 | Valid method leading to \(h\) in terms of \(r\) Co |
| \(\rightarrow V = \pi r^2h \rightarrow V = 12\pi r^2 - 2\pi r^3\) | M1 | Use of vol formula with some \(h=f(r)\) |
| (ii) \(\frac{dV}{dr} = 24\pi r - 6\pi r^2\) | M1 A1 | Attempt at differentiation, (2 or \(\pi\) or \(2\pi\) missing, loses this A1 only) |
| \(= 0\) when \(r = 4 \rightarrow V = 64\pi\) (or 201) | M1 A1 | Setting his differential to 0. co. |
(i) Similar triangles or trig ($\tan = \text{opp}/\text{hyp}$)
$\frac{6}{12} = \frac{r}{12-h} \rightarrow h = 12 - 2r$ | M1 A1 | Valid method leading to $h$ in terms of $r$ Co
$\rightarrow V = \pi r^2h \rightarrow V = 12\pi r^2 - 2\pi r^3$ | M1 | Use of vol formula with some $h=f(r)$ | [3]
(ii) $\frac{dV}{dr} = 24\pi r - 6\pi r^2$ | M1 A1 | Attempt at differentiation, (2 or $\pi$ or $2\pi$ missing, loses this A1 only)
$= 0$ when $r = 4 \rightarrow V = 64\pi$ (or 201) | M1 A1 | Setting his differential to 0. co. | [4](i) Express $h$ in terms of $r$ and hence show that the volume, $V \mathrm {~cm} ^ { 3 }$, of the cylinder is given by
$$V = 12 \pi r ^ { 2 } - 2 \pi r ^ { 3 }$$
(ii) Given that $r$ varies, find the stationary value of $V$.
\hfill \mbox{\textit{CAIE P1 2005 Q5 [7]}}