OCR MEI C3 2011 January — Question 3 5 marks

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
Year2011
SessionJanuary
Marks5
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Mark schemeDownload PDF ↗
TopicConnected Rates of Change
TypeBalloon or expanding shape
DifficultyStandard +0.3 This is a straightforward related rates problem requiring differentiation of A = πr² with respect to time, then substituting given values. It's slightly above average difficulty because it requires understanding the chain rule and setting up the relationship dA/dt = 2πr(dr/dt), but follows a standard template that students practice extensively in C3.
Spec1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
The area of a circular stain is growing at a rate of \(1 \text{ mm}^2\) per second. Find the rate of increase of its radius at an instant when its radius is \(2\) mm. [5]
AnswerMarks Guidance
\(A = \pi r^2\)M1A1, A1 [5] M1A0 if incorrect notation, e.g. \(dy/dx\), \(dr/dA\), if seen. \(2r\) is M1A0 must be \(dA/dr\) (soi) and \(dA/dr\) any correct form stated with relevant variables , e.g. \(\frac{dr}{dA}, \frac{dA}{dr}, \frac{dr}{dt}, \frac{dr}{dA}, \frac{dt}{dA}\), etc.
\(\Rightarrow \quad \frac{dA}{dr} = 2\pi r\)
When \(r = 2\), \(\frac{dA}{dr} = 4\pi\), \(\frac{dA}{dr} = 1\)
\(\frac{dA}{dr} \div \frac{dA}{dr} \div \frac{dr}{dt}\)
\(\Rightarrow \quad 1 = 4\pi \frac{dr}{dt}\)M1 chain rule (o.e)
\(\Rightarrow \quad dr/dt = 1/4\pi = 0.0796\) (mm/s)A1 [5] cao: 0.08 or better condone truncation 
$A = \pi r^2$ | M1A1, A1 [5] | M1A0 if incorrect notation, e.g. $dy/dx$, $dr/dA$, if seen. $2r$ is M1A0 must be $dA/dr$ (soi) and $dA/dr$ any correct form stated with relevant variables , e.g. $\frac{dr}{dA}, \frac{dA}{dr}, \frac{dr}{dt}, \frac{dr}{dA}, \frac{dt}{dA}$, etc.

$\Rightarrow \quad \frac{dA}{dr} = 2\pi r$ | |

When $r = 2$, $\frac{dA}{dr} = 4\pi$, $\frac{dA}{dr} = 1$ | |

$\frac{dA}{dr} \div \frac{dA}{dr} \div \frac{dr}{dt}$ | |

$\Rightarrow \quad 1 = 4\pi \frac{dr}{dt}$ | M1 | chain rule (o.e) | allow $\frac{1}{4\pi}$ but mark final answer

$\Rightarrow \quad dr/dt = 1/4\pi = 0.0796$ (mm/s) | A1 [5] | cao: 0.08 or better condone truncation

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The area of a circular stain is growing at a rate of $1 \text{ mm}^2$ per second. Find the rate of increase of its radius at an instant when its radius is $2$ mm.
[5]

\hfill \mbox{\textit{OCR MEI C3 2011 Q3 [5]}}
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