OCR MEI C3 — Question 2 5 marks

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
Marks5
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicVariable acceleration (1D)
TypeRelated rates with spheres, circles, and cubes
DifficultyStandard +0.3 This is a straightforward related rates problem requiring the chain rule (dV/dt = dV/dr × dr/dt) with a given formula. Students substitute r=8 into the differentiated volume formula and solve for dr/dt. It's slightly easier than average as it's a standard textbook exercise with clear setup and direct application of a single technique.
Spec1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
2 A spherical balloon of radius \(r \mathrm {~cm}\) has volume \(V \mathrm {~cm} ^ { 3 }\), where \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\). The balloon is inflated at a constant rate of \(10 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of \(r\) when \(r = 8\).
Question 2:
AnswerMarks Guidance
\(\frac{dV}{dr} = 4\pi r^2\)B1 or \(12\pi r^2/3\), condone \(\frac{dr}{dV}\), \(\frac{dV}{dR}\)
\(\frac{dV}{dt} = 10\)B1 Condone use of other letters for \(t\)
\(\frac{dV}{dt} = \left(\frac{dV}{dr}\right)\left(\frac{dr}{dt}\right)\)M1 a correct chain rule soi; o.e. e.g. \(\frac{dr}{dt} = \left(\frac{dr}{dV}\right)\left(\frac{dV}{dt}\right)\)
\(10 = 4\pi r^2 \cdot \frac{dr}{dt}\)A1 (soi) must be correct
\(\frac{dr}{dt} = 0.0124\) cm s\(^{-1}\)A1 0.012 or better or \(\frac{10}{256\pi}\) or \(\frac{5}{128\pi}\); mark final answer
[5]
Question 2:

$\frac{dV}{dr} = 4\pi r^2$ | B1 | or $12\pi r^2/3$, condone $\frac{dr}{dV}$, $\frac{dV}{dR}$

$\frac{dV}{dt} = 10$ | B1 | Condone use of other letters for $t$

$\frac{dV}{dt} = \left(\frac{dV}{dr}\right)\left(\frac{dr}{dt}\right)$ | M1 | a correct chain rule soi; o.e. e.g. $\frac{dr}{dt} = \left(\frac{dr}{dV}\right)\left(\frac{dV}{dt}\right)$

$10 = 4\pi r^2 \cdot \frac{dr}{dt}$ | A1 | (soi) must be correct

$\frac{dr}{dt} = 0.0124$ cm s$^{-1}$ | A1 | 0.012 or better or $\frac{10}{256\pi}$ or $\frac{5}{128\pi}$; mark final answer

[5]
2 A spherical balloon of radius $r \mathrm {~cm}$ has volume $V \mathrm {~cm} ^ { 3 }$, where $V = \frac { 4 } { 3 } \pi r ^ { 3 }$. The balloon is inflated at a constant rate of $10 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }$. Find the rate of increase of $r$ when $r = 8$.

\hfill \mbox{\textit{OCR MEI C3  Q2 [5]}}
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