Level 3 Differentiation Walkthrough

2016 NCEA Level 3 Differentiation Question 2(d)

2016 Paper

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Question

A large spherical helium balloon is being inflated at a constant rate of

\[4800\text{ cm}^3\text{ s}^{-1}.\]

At what rate is the radius of the balloon increasing when the volume of the balloon is \(288000\pi\text{ cm}^3\)?

You must use calculus and show any derivatives that you need to find when solving this problem.

First walkthrough idea

Focus to try first

Use the stated volume to find the radius at that instant, then differentiate the sphere-volume formula with respect to time.

Step 1

Find the radius at the stated volume

The related-rates equation needs the radius at this particular instant.

Show the first step’s working
\[V=\frac43\pi r^3\] \[288000\pi=\frac43\pi r^3\] \[r^3=216000\] \[r=60\text{ cm}.\]

Walkthrough overview

What this question practises

This 2016 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Related rates for the radius of an inflating sphere.

This is Question 2(d) from the 2016 NCEA Level 3 Differentiation paper for AS91578 — Apply differentiation methods in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91578.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise related rates for the radius of an inflating sphere. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Differentiate with respect to time consistently, then include the correct units and contextual interpretation.

Continue practising

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