Level 3 Differentiation Walkthrough

2025 NCEA Level 3 Differentiation Question 3(e)

2025 Paper

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Question

The diagram below shows part of the symmetrical graph \(y^2=16x-x^2\).

A B C D y² = 16x - x² x y

A rectangle \(ABCD\) is drawn inside the curve with its vertices \(B\) and \(C\) lying on the curve.

Find the length \(AD\) so that the rectangle has its maximum area.

You can assume that the value found is a maximum.

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

The diagram is shown in its initial state. JavaScript adds any interactive controls and later walkthrough visuals.

First walkthrough idea

Hint to try first

Let \(A=(x,0)\), so by symmetry \(D\) is at \(x=16-x\).

Step 1

Write the area function

Width times height gives \((16-2x)\sqrt{16x-x^2}\).

Show the first step’s working

Width times height gives \((16-2x)\sqrt{16x-x^2}\).

Key result

\[ \left(16 - 2 x\right) \sqrt{16 x - x^{2}} \]

Walkthrough overview

What this question practises

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

Method: Maximising the area of a rectangle inside a curve.

This is Question 3(e) 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.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise maximising the area of a rectangle inside a curve. 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 3(d)