CAIE P1 2012 November — Question 3 4 marks

Exam BoardCAIE
ModuleP1 (Pure Mathematics 1)
Year2012
SessionNovember
Marks4
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Mark schemeDownload PDF ↗
TopicApplied differentiation
TypeConnected rates of change
DifficultyModerate -0.8 This is a straightforward related rates problem requiring only the chain rule applied to A = πr². Given dr/dt = 3 m/h at r = 50 m, students differentiate to get dA/dt = 2πr(dr/dt) and substitute values. It's a single-step application of a standard technique with no conceptual complications, making it easier than average but not trivial since it requires understanding the relationship between rates.
Spec1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
3 An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is 50 m and is increasing at a rate of 3 metres per hour. Find the rate at which the area of the oil is increasing at midday.
AnswerMarks Guidance
\(A = \pi r^2 \to (\frac{dA}{dr}) = 2\pi r\)B1
\(\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}\) usedM1
\(\frac{dr}{dt} = 3\) soiB1
\(300\pi\) (or 942)A1 [4] 
$A = \pi r^2 \to (\frac{dA}{dr}) = 2\pi r$ | B1 |
$\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}$ used | M1 |
$\frac{dr}{dt} = 3$ soi | B1 |
$300\pi$ (or 942) | A1 | [4]
3 An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is 50 m and is increasing at a rate of 3 metres per hour. Find the rate at which the area of the oil is increasing at midday.

\hfill \mbox{\textit{CAIE P1 2012 Q3 [4]}}
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