Level 3 Differentiation Walkthrough

2019 NCEA Level 3 Differentiation Question 1(e)

2019 Paper

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Question

Scanned exam prompt asking for the rate of increase of a sphere's volume from its radius and surface-area rate.

First walkthrough idea

Focus to try first

Related rates connecting surface area, radius, and volume of a sphere.

Step 1

Connect the rates

Use the chain rule to connect volume to surface area through the radius.

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

Walkthrough overview

What this question practises

This 2019 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

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

This is Question 1(e) from the 2019 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 surface area 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

← Question 1(d) Question 2(a) →