| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Optimization with constraint |
| Difficulty | Standard +0.3 This is a standard optimization problem requiring expressing surface area in terms of one variable using a constraint, then differentiating and finding stationary points. The algebra is straightforward (substituting h=250/r² into S=2πr²+2πrh), differentiation is routine, and determining nature via second derivative is a textbook technique. Slightly above average difficulty due to multi-step nature but requires no novel insight. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow S = 2\pi r^2 + \frac{500\pi}{r}\) | M1 M1 | Makes \(h\) the subject. \(\pi r^2 h\) must be right. Ans given – check all formulae |
| [2] | ||
| (ii) \(\frac{dS}{dr} = 4\pi r - \frac{500\pi}{r^2}\) | B1 B1 | B1 for each term |
| \(0\) when \(r^3 = 125 \rightarrow r = 5\) → \(S = 150\pi\) | M1 A1 | Sets differential to 0 + attempt at soln |
| [4] | ||
| (iii) \(\frac{d^2S}{dr^2} = 4\pi + \frac{1000\pi}{r^3}\). This is positive → Minimum | M1 A1 | Any valid method. 2nd differential must be correct – no need for numerical answer or correct \(r\) |
| [2] |
(i) $\pi r^2 h = 250\pi \rightarrow h = \frac{250}{r^2}$
$\rightarrow S = 2\pi rh + 2\pi r^2$
$\rightarrow S = 2\pi r^2 + \frac{500\pi}{r}$ | M1 M1 | Makes $h$ the subject. $\pi r^2 h$ must be right. Ans given – check all formulae
[2] |
(ii) $\frac{dS}{dr} = 4\pi r - \frac{500\pi}{r^2}$ | B1 B1 | B1 for each term
$0$ when $r^3 = 125 \rightarrow r = 5$ → $S = 150\pi$ | M1 A1 | Sets differential to 0 + attempt at soln
[4] |
(iii) $\frac{d^2S}{dr^2} = 4\pi + \frac{1000\pi}{r^3}$. This is positive → Minimum | M1 A1 | Any valid method. 2nd differential must be correct – no need for numerical answer or correct $r$
[2] |
---8 The volume of a solid circular cylinder of radius $r \mathrm {~cm}$ is $250 \pi \mathrm {~cm} ^ { 3 }$.\\
(i) Show that the total surface area, $S \mathrm {~cm} ^ { 2 }$, of the cylinder is given by
$$S = 2 \pi r ^ { 2 } + \frac { 500 \pi } { r }$$
(ii) Given that $r$ can vary, find the stationary value of $S$.\\
(iii) Determine the nature of this stationary value.
\hfill \mbox{\textit{CAIE P1 2013 Q8 [8]}}