Level 3 Differentiation Walkthrough

2017 NCEA Level 3 Differentiation Question 2(e)

2017 Paper

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Question

A rectangle is inscribed in a semicircle of radius \(r\), as shown below.

Rectangle inscribed in a semicircle A semicircle of radius r is centred on its diameter. A symmetric rectangle has half-width x and height y, with both upper corners on the arc. −r −x x r y width = 2x

Show that the maximum possible area of such a rectangle occurs when

\[x=\frac{r}{\sqrt2}.\]

You do not need to prove that your solution gives the maximum area.

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 the circle equation to express the rectangle's height in terms of its half-width, then differentiate the resulting area function.

Step 1

Relate the height and half-width

One upper corner, the centre, and a radius form a right triangle.

Show the first step’s working

Let \(x\) be half the rectangle's width and \(y\) its height. Then

\[x^2+y^2=r^2\] \[y^2=r^2-x^2\] \[y=\sqrt{r^2-x^2}.\]

Here \(0\le x\le r\), and the positive square root is used because \(y\) is a height.

Walkthrough overview

What this question practises

This 2017 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Maximising the area of a rectangle in a semicircle.

This is Question 2(e) from the 2017 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 in a semicircle. 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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