OCR MEI C2 2008 January — Question 10 12 marks

Exam BoardOCR MEI
ModuleC2 (Core Mathematics 2)
Year2008
SessionJanuary
Marks12
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicApplied differentiation
TypeOptimise 3D shape dimensions
DifficultyModerate -0.3 This is a standard C2 optimization problem with clear scaffolding: express constraint, derive surface area formula (given), differentiate, find minimum. All steps are routine applications of basic differentiation and algebra with no novel problem-solving required. Slightly easier than average due to heavy guidance.
Spec1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives
10 \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-3_501_493_1434_826} \captionsetup{labelformat=empty} \caption{Fig. 10}
\end{figure} Fig. 10 shows a solid cuboid with square base of side \(x \mathrm {~cm}\) and height \(h \mathrm {~cm}\). Its volume is \(120 \mathrm {~cm} ^ { 3 }\).
  1. Find \(h\) in terms of \(x\). Hence show that the surface area, \(A \mathrm {~cm} ^ { 2 }\), of the cuboid is given by \(A = 2 x ^ { 2 } + \frac { 480 } { x }\).
  2. Find \(\frac { \mathrm { d } A } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } A } { \mathrm {~d} x ^ { 2 } }\).
  3. Hence find the value of \(x\) which gives the minimum surface area. Find also the value of the surface area in this case.
i
\(h = 120/x^2\)
\(A = 2x^2 + 4xh\) o.e.
AnswerMarks Guidance
completion to given answerB1, M1, A1 at least one interim step shown
ii
\(A' = 4x - 480/x^2\) o.e.
AnswerMarks Guidance
\(A'' = 4 + 960/x^3\)2, 2 1 for \(kx^2\) o.e. included; ft their \(A'\) only if \(kx^2\) seen; 1 if one error
iii
use of \(A' = 0\)
AnswerMarks Guidance
\(x = \sqrt[3]{120}\) or \(4.9(3...)\)M1, A1 —
Test using \(A'\) or \(A''\) to confirm minimum
Substitution of their x in A
AnswerMarks Guidance
\(A = 145.9\) to 146T1, M1, A1 Dependent on previous M1 
**i**

$h = 120/x^2$
$A = 2x^2 + 4xh$ o.e.
completion to given answer | B1, M1, A1 | at least one interim step shown | 3

**ii**

$A' = 4x - 480/x^2$ o.e.
$A'' = 4 + 960/x^3$ | 2, 2 | 1 for $kx^2$ o.e. included; ft their $A'$ only if $kx^2$ seen; 1 if one error | 4

**iii**

use of $A' = 0$
$x = \sqrt[3]{120}$ or $4.9(3...)$ | M1, A1 | — | —
Test using $A'$ or $A''$ to confirm minimum
Substitution of their x in A
$A = 145.9$ to 146 | T1, M1, A1 | Dependent on previous M1 | 5

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10

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{15872003-2e41-47e9-a5bd-34e533768f8a-3_501_493_1434_826}
\captionsetup{labelformat=empty}
\caption{Fig. 10}
\end{center}
\end{figure}

Fig. 10 shows a solid cuboid with square base of side $x \mathrm {~cm}$ and height $h \mathrm {~cm}$. Its volume is $120 \mathrm {~cm} ^ { 3 }$.\\
(i) Find $h$ in terms of $x$. Hence show that the surface area, $A \mathrm {~cm} ^ { 2 }$, of the cuboid is given by $A = 2 x ^ { 2 } + \frac { 480 } { x }$.\\
(ii) Find $\frac { \mathrm { d } A } { \mathrm {~d} x }$ and $\frac { \mathrm { d } ^ { 2 } A } { \mathrm {~d} x ^ { 2 } }$.\\
(iii) Hence find the value of $x$ which gives the minimum surface area. Find also the value of the surface area in this case.

\hfill \mbox{\textit{OCR MEI C2 2008 Q10 [12]}}
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