Level 3 Differentiation Walkthrough

2018 NCEA Level 3 Differentiation Question 3(e)

2018 Paper

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Question

Symmetric wire inside a ten centimetre square A central vertical wire of length x joins two branch points. Four equal diagonal wires run from the branch points to the corners of a ten centimetre square. Dashed vertical and horizontal lines show symmetry. 10 cm 10 cm x y vertical symmetry horizontal symmetry

The shape is made from wire and has both vertical and horizontal lines of symmetry. Its four ends are at the vertices of a square with side length \(10\text{ cm}\).

The length of the piece of wire through the centre is \(x\text{ cm}\). Find the value(s) of \(x\) that enables the shape to be made with the minimum length of wire.

You do not need to prove that the length is a minimum. You must use calculus and show any derivatives that you need to find when solving this problem.

First walkthrough idea

Focus to try first

Use symmetry and Pythagoras to express one diagonal length in terms of \(x\), then minimise the total length.

Step 1

Find one diagonal length

Each diagonal has horizontal displacement \(5\) and vertical displacement \((10-x)/2\).

Show the first step’s working
\[y^2=\left(\frac{10-x}{2}\right)^2+5^2\] \[y^2=\frac{x^2}{4}-5x+50\] \[y=\sqrt{\frac{x^2}{4}-5x+50}\]

Walkthrough overview

What this question practises

This 2018 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Building and minimising a symmetric wire-length model.

This is Question 3(e) from the 2018 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 building and minimising a symmetric wire-length model. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 3(d)