| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Marks | 11 |
| 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 standard AS-level optimization problem combining surface area/volume formulas with basic calculus. Part (a) requires algebraic manipulation to express h in terms of x from the surface area constraint, then substituting into the volume formula. Part (b) involves routine differentiation, solving dV/dx=0, and using the second derivative test. Part (c) is simple substitution. While multi-step, each component uses well-practiced techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.02z Models in context: use functions in modelling1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| \(V = 60x - \frac{4x^3}{3}\) | M1, A1, M1, A1 | Attempting an expression in terms of \(x\) and \(y\) for the total surface area; Correct expression equated to 180; Substitute \(h\) into \(V = 2x^2h\) to form an expression in terms of \(x\) only; Correct solution only |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = 3.87\) (to 3 significant figures) | B1, M1, A1, M1, A1 | Finding the first derivative (does not have to be correct); Equate their \(\frac{dV}{dx}\) to 0 and solve for \(x^2\) or \(x\); Correct \(x\) values; Finding the second derivative and considering the sign; Correct reason and conclusion |
| Answer | Marks | Guidance |
|---|---|---|
| \(V = 155\) cm³ (to the nearest cm³) | M1, A1 | Substitutes their \(x\) value into the given expression for \(V\); Correct answer only |
### Part a:
$V = 60x - \frac{4x^3}{3}$ | M1, A1, M1, A1 | Attempting an expression in terms of $x$ and $y$ for the total surface area; Correct expression equated to 180; Substitute $h$ into $V = 2x^2h$ to form an expression in terms of $x$ only; Correct solution only
### Part b:
$x = 3.87$ (to 3 significant figures) | B1, M1, A1, M1, A1 | Finding the first derivative (does not have to be correct); Equate their $\frac{dV}{dx}$ to 0 and solve for $x^2$ or $x$; Correct $x$ values; Finding the second derivative and considering the sign; Correct reason and conclusion
### Part c:
$V = 155$ cm³ (to the nearest cm³) | M1, A1 | Substitutes their $x$ value into the given expression for $V$; Correct answer only\includegraphics{figure_2}
Figure 2 shows a solid cuboid $ABCDEFGH$.
$AB = x$ cm, $BC = 2x$ cm, $AE = h$ cm
The total surface area of the cuboid is 180 cm$^2$.
The volume of the cuboid is $V$ cm$^3$.
\begin{enumerate}[label=(\alph*)]
\item Show that $V = 60x - \frac{4x^3}{3}$ [4]
\end{enumerate}
Given that $x$ can vary,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item use calculus to find, to 3 significant figures, the value of $x$ for which $V$ is a maximum. Justify that this value of $x$ gives a maximum value of $V$. [5]
\item Find the maximum value of $V$, giving your answer to the nearest cm$^3$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel AS Paper 1 Q8 [11]}}