Level 3 Differentiation Walkthrough

2021 NCEA Level 3 Differentiation Question 1(e)

2021 Paper

← Back to paper

Question

A cone has a height of \(3\text{ m}\) and a radius of \(1.5\text{ m}\).

A cylinder is inscribed in the cone, as shown in the diagram below.

3 m 1.5 m h r

The base of the cylinder has the same centre as the base of the cone.

Prove that the maximum volume of the cylinder is \(\pi\text{ m}^3\).

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

First walkthrough idea

Focus to try first

First express the cylinder height \(h\) in terms of its radius \(r\). Then make a one-variable volume function.

Step 1

Relate h and r

Use similar triangles, or the straight line from \((0,3)\) to \((1.5,0)\).

Show the first step’s working

When \(r=0\), the available height is \(3\). When \(r=1.5\), the height is \(0\). This gives a straight-line relationship.

\[ h=3-2r \]

Walkthrough overview

What this question practises

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

Method: Cone and cylinder optimisation with a maximum-volume proof.

This is Question 1(e) 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.

Page updated .

Learning summary

Review the method, not only the answer

This walkthrough helps you practise cone and cylinder optimisation with a maximum-volume proof. 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

← Question 1(d) Question 2(a) →