CAIE P1 2004 June — Question 8 10 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2004
SessionJune
Marks10
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicChain Rule
TypeOptimization with constraint
DifficultyStandard +0.3 This is a standard optimization problem requiring constraint manipulation, area formula derivation, and basic differentiation. While it involves multiple steps, each step follows routine procedures (expressing one variable in terms of another, differentiating a polynomial, and using the second derivative test). The perimeter constraint is straightforward, and the calculus is elementary with no conceptual challenges beyond typical A-level expectations.
Spec1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.07p Points of inflection: using second derivative
8 \includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-4_543_511_264_817} The diagram shows a glass window consisting of a rectangle of height \(h \mathrm {~m}\) and width \(2 r \mathrm {~m}\) and a semicircle of radius \(r \mathrm {~m}\). The perimeter of the window is 8 m .
  1. Express \(h\) in terms of \(r\).
  2. Show that the area of the window, \(A \mathrm {~m} ^ { 2 }\), is given by $$A = 8 r - 2 r ^ { 2 } - \frac { 1 } { 2 } \pi r ^ { 2 } .$$ Given that \(r\) can vary,
  3. find the value of \(r\) for which \(A\) has a stationary value,
  4. determine whether this stationary value is a maximum or a minimum.
(i) \(2h + 2r + \pi r = 8\) → \(h = 4 - r - \frac{1}{2}\pi r\)
AnswerMarks
M1 A1 [2]Reasonable attempt at linking 4 lengths + correct formula for \(\frac{1}{3}C\) or \(C\); Co in any form with \(h\) subject.
(ii) \(A = 2rh + \pi r^2 \to A = r(8-2r-\pi r) + \frac{1}{2}\pi r^2\) → \(A = 8r - 2r^2 - \frac{1}{2}\pi r^2\)
AnswerMarks
M1 A1 [2]Adds rectangle + \(\frac{1}{3}\)circle (eqn on own ok); Co beware fortuitous answers (ans given)
(iii) \(dA/dr = 8 - 4r - \pi r = 0\) when \(r = 1.12\) (or \(8/(4+\pi)\))
AnswerMarks
M1 A1 DM1 A1 [4]Knowing to differentiate + some attempt; Setting his \(dA/dr\) to 0, Decimal or exact ok.
(iv) \(d^2A/dr^2 = -4 - \pi\) This is negative → Maximum
AnswerMarks
M1 A1 [2]Looks at \(2^{nd}\) differential or other valid complete method; Correct deduction but needs \(d^2A/dr^2\) correct. 
**(i)** $2h + 2r + \pi r = 8$ → $h = 4 - r - \frac{1}{2}\pi r$
| M1 A1 [2] | Reasonable attempt at linking 4 lengths + correct formula for $\frac{1}{3}C$ or $C$; Co in any form with $h$ subject. |

**(ii)** $A = 2rh + \pi r^2 \to A = r(8-2r-\pi r) + \frac{1}{2}\pi r^2$ → $A = 8r - 2r^2 - \frac{1}{2}\pi r^2$
| M1 A1 [2] | Adds rectangle + $\frac{1}{3}$circle (eqn on own ok); Co beware fortuitous answers (ans given) |

**(iii)** $dA/dr = 8 - 4r - \pi r = 0$ when $r = 1.12$ (or $8/(4+\pi)$)
| M1 A1 DM1 A1 [4] | Knowing to differentiate + some attempt; Setting his $dA/dr$ to 0, Decimal or exact ok. |

**(iv)** $d^2A/dr^2 = -4 - \pi$ This is negative → Maximum
| M1 A1 [2] | Looks at $2^{nd}$ differential or other valid complete method; Correct deduction but needs $d^2A/dr^2$ correct. |

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8\\
\includegraphics[max width=\textwidth, alt={}, center]{22a31966-4433-4d7d-8a75-bcd536acfa24-4_543_511_264_817}

The diagram shows a glass window consisting of a rectangle of height $h \mathrm {~m}$ and width $2 r \mathrm {~m}$ and a semicircle of radius $r \mathrm {~m}$. The perimeter of the window is 8 m .\\
(i) Express $h$ in terms of $r$.\\
(ii) Show that the area of the window, $A \mathrm {~m} ^ { 2 }$, is given by

$$A = 8 r - 2 r ^ { 2 } - \frac { 1 } { 2 } \pi r ^ { 2 } .$$

Given that $r$ can vary,\\
(iii) find the value of $r$ for which $A$ has a stationary value,\\
(iv) determine whether this stationary value is a maximum or a minimum.

\hfill \mbox{\textit{CAIE P1 2004 Q8 [10]}}
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