Level 3 Differentiation Walkthrough

2025 NCEA Level 3 Differentiation Question 3(c)

2025 Paper

← Back to paper

Question

A solid spherical lump of ice is melting while maintaining its spherical shape.

At the instant when the radius is \(6\) cm, the radius of the ball of ice is decreasing by \(0.05\) cm s\(^{-1}\).

Find the rate at which the volume of the spherical ice ball is decreasing at the instant when the radius is \(6\) cm.

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

First walkthrough idea

Hint to try first

Use the sphere volume formula \(V=\frac{4}{3}\pi r^3\).

Step 1

Link the rates

This is the chain rule written for related rates.

Show the first step’s working

This is the chain rule written for related rates.

Key result

\[ \frac{dV}{dt}=\frac{dV}{dr}\cdot\frac{dr}{dt} \]

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Related rates for the volume of a sphere.

This is Question 3(c) from the 2025 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.

Page updated .

Learning summary

Review the method, not only the answer

This walkthrough helps you practise related rates for the volume 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 3(b) Question 3(d) →