Edexcel C4 2010 January — Question 6 5 marks

Exam BoardEdexcel
ModuleC4 (Core Mathematics 4)
Year2010
SessionJanuary
Marks5
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Mark schemeDownload PDF ↗
TopicVariable acceleration (1D)
TypeRelated rates with spheres, circles, and cubes
DifficultyStandard +0.3 This is a standard related rates problem requiring the chain rule (dA/dt = dA/dr × dr/dt) with the circle area formula A = πr². Students must rearrange to find dr/dt and substitute given values. It's slightly above average difficulty due to requiring conceptual understanding of implicit differentiation and careful algebraic manipulation, but follows a well-practiced C4 template with no novel insight required.
Spec1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
6. The area \(A\) of a circle is increasing at a constant rate of \(1.5 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\). Find, to 3 significant figures, the rate at which the radius \(r\) of the circle is increasing when the area of the circle is \(2 \mathrm {~cm} ^ { 2 }\).
(5)
AnswerMarks Guidance
\(\frac{dA}{dt} = 1.5\) then \(A = \pi r^2 \Rightarrow \frac{dA}{dr} = 2\pi r\)B1; B1
When \(A = 2\): \(2 = \pi r^2 \Rightarrow r = \sqrt{\frac{2}{\pi}}\) (= 0.797884...)M1
\(\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}\) then \(1.5 = 2\pi r \frac{dr}{dt}\) then \(\frac{dr}{dt} = \frac{1.5}{2\pi\sqrt{\frac{2}{\pi}}} \approx 0.299\)M1; awrt 0.299 A1 [5] 
| $\frac{dA}{dt} = 1.5$ then $A = \pi r^2 \Rightarrow \frac{dA}{dr} = 2\pi r$ | B1; B1 |

When $A = 2$: $2 = \pi r^2 \Rightarrow r = \sqrt{\frac{2}{\pi}}$ (= 0.797884...) | M1 |

$\frac{dA}{dt} = \frac{dA}{dr} \times \frac{dr}{dt}$ then $1.5 = 2\pi r \frac{dr}{dt}$ then $\frac{dr}{dt} = \frac{1.5}{2\pi\sqrt{\frac{2}{\pi}}} \approx 0.299$ | M1; awrt 0.299 A1 | **[5]**

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6. The area $A$ of a circle is increasing at a constant rate of $1.5 \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }$. Find, to 3 significant figures, the rate at which the radius $r$ of the circle is increasing when the area of the circle is $2 \mathrm {~cm} ^ { 2 }$.\\
(5)\\

\hfill \mbox{\textit{Edexcel C4 2010 Q6 [5]}}
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