WJEC Unit 3 Specimen — Question 2 3 marks

Exam BoardWJEC
ModuleUnit 3 (Unit 3)
SessionSpecimen
Marks3
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Mark schemeDownload PDF ↗
TopicConnected Rates of Change
TypeSphere: radius rate from volume rate
DifficultyStandard +0.3 This is a standard related rates problem requiring differentiation of the sphere volume formula V = (4/3)πr³, then substituting given values. It's slightly above average difficulty as it requires understanding implicit differentiation with respect to time and careful algebraic manipulation, but follows a well-practiced technique taught in all A-level courses.
Spec1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates
Air is pumped into a spherical balloon at the rate of 250 cm\(^3\) per second. When the radius of the balloon is 15 cm, calculate the rate at which the radius is increasing, giving your answer to three decimal places [3]
AnswerMarks Guidance
\(V = \frac{4}{3}\pi r^3\)
\(\frac{dV}{dt} = 3 \times \frac{4}{3}\pi r^2 \frac{dr}{dt}\)B1
\(4\pi \times 15^2 \times \frac{dr}{dt} = 250\)M1 (Substitution of data)
\(\frac{dr}{dt} = \frac{250}{900\pi} \approx 0.088\) (cm/second)A1
Total: [3]
$V = \frac{4}{3}\pi r^3$ | |

$\frac{dV}{dt} = 3 \times \frac{4}{3}\pi r^2 \frac{dr}{dt}$ | B1 |

$4\pi \times 15^2 \times \frac{dr}{dt} = 250$ | M1 | (Substitution of data)

$\frac{dr}{dt} = \frac{250}{900\pi} \approx 0.088$ (cm/second) | A1 |

**Total: [3]**
Air is pumped into a spherical balloon at the rate of 250 cm$^3$ per second. When the radius of the balloon is 15 cm, calculate the rate at which the radius is increasing, giving your answer to three decimal places [3]

\hfill \mbox{\textit{WJEC Unit 3  Q2 [3]}}
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