| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Optimization with constraint |
| Difficulty | Standard +0.3 This is a standard optimization problem with a linear constraint. Part (i) is straightforward substitution, part (ii) requires basic differentiation and the second derivative test, and part (iii) is simple evaluation. The problem follows a familiar textbook pattern with no novel insights required, making it slightly easier than average for A-level. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(V = \frac{1}{3}\pi r^2(18-r) = 6\pi r^2 - \frac{1}{3}\pi r^3\) | B1 | AG |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\frac{dV}{dr} = 12\pi r - \pi r^2 = 0\) | M1 | Differentiate and set \(= 0\) |
| \(\pi r(12 - r) = 0 \rightarrow r = 12\) | A1 | |
| \(\frac{d^2V}{dr^2} = 12\pi - 2\pi r\) | M1 | |
| Sub \(r = 12 \rightarrow 12\pi - 24\pi = -12\pi \rightarrow\) MAX | A1 | AG |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Sub \(r = 12\), \(h = 6 \rightarrow\) Max \(V = 288\pi\) or \(905\) | B1 | |
| Total: 1 |
## Question 4(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $V = \frac{1}{3}\pi r^2(18-r) = 6\pi r^2 - \frac{1}{3}\pi r^3$ | B1 | AG |
| **Total: 1** | | |
---
## Question 4(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dV}{dr} = 12\pi r - \pi r^2 = 0$ | M1 | Differentiate and set $= 0$ |
| $\pi r(12 - r) = 0 \rightarrow r = 12$ | A1 | |
| $\frac{d^2V}{dr^2} = 12\pi - 2\pi r$ | M1 | |
| Sub $r = 12 \rightarrow 12\pi - 24\pi = -12\pi \rightarrow$ MAX | A1 | AG |
| **Total: 4** | | |
---
## Question 4(iii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Sub $r = 12$, $h = 6 \rightarrow$ Max $V = 288\pi$ or $905$ | B1 | |
| **Total: 1** | | |4 Machines in a factory make cardboard cones of base radius $r \mathrm {~cm}$ and vertical height $h \mathrm {~cm}$. The volume, $V \mathrm {~cm} ^ { 3 }$, of such a cone is given by $V = \frac { 1 } { 3 } \pi r ^ { 2 } h$. The machines produce cones for which $h + r = 18$.\\
(i) Show that $V = 6 \pi r ^ { 2 } - \frac { 1 } { 3 } \pi r ^ { 3 }$.\\
(ii) Given that $r$ can vary, find the non-zero value of $r$ for which $V$ has a stationary value and show that the stationary value is a maximum.\\
(iii) Find the maximum volume of a cone that can be made by these machines.\\
\hfill \mbox{\textit{CAIE P1 2017 Q4 [6]}}