Level 2 Calculus Walkthrough

2025 NCEA Level 2 Calculus Question 3(c)

2025 Paper — Maximise the volume of a lidless cuboid

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Question

A lidless rectangular cuboid has surface area \(4.32\text{ m}^2\).

Its width is twice its height, and the volume is to be maximised.

Use calculus to find the maximum volume and prove that it is a maximum.

Start from the lidless surface-area formula, use \(w=2h\) to write \(l\) in terms of \(h\), then form a volume function, maximise it, and use the second derivative to prove it is a maximum.

First walkthrough idea

Tip to try first

Because the container is lidless, the material covers the base and four side faces only.

Step 1

Write the surface-area constraint

Reveal the explanation and working for this step.

Show the first step’s working

Account for the missing lid when writing the surface-area constraint.

Key result

\[ lw+2lh+2wh=4.32 \]

A lidless box has one base and four side faces, so the top is not included.

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91262 — Apply calculus methods in solving problems.

Method: Using a surface-area constraint, writing volume in one variable, maximizing it, and proving the maximum.

This is Question 3(c) from the 2025 NCEA Level 2 Calculus paper for AS91262 — Apply calculus 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 AS91262.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise using a surface-area constraint, writing volume in one variable, maximizing it, and proving the maximum. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Finding a stationary value is only part of an optimisation argument; justify that it is the required maximum and respect the domain.

Continue practising

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