Level 3 Differentiation Walkthrough

2021 NCEA Level 3 Differentiation Question 2(d)

2021 Paper

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Question

The volume of a spherical balloon is increasing at a constant rate of \(60\text{ cm}^3\) per second.

Find the rate of increase of the radius when the radius is \(15\text{ cm}\).

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

First walkthrough idea

Focus to try first

Connect \(\frac{dV}{dt}\) and \(\frac{dr}{dt}\) using \(\frac{dV}{dt}=\frac{dV}{dr}\frac{dr}{dt}\).

Step 1

Write what is given

The volume rate is already a derivative with respect to time.

Show the first step’s working
\[ \frac{dV}{dt}=60 \]

We need \(\frac{dr}{dt}\) when \(r=15\).

Walkthrough overview

What this question practises

This 2021 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Related rates for the volume and radius of a sphere.

This is Question 2(d) from the 2021 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 volume and radius of a 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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