Level 3 Differentiation Walkthrough

2018 NCEA Level 3 Differentiation Question 3(c)

2018 Paper

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Question

The diagram shows the graph of

\[y=15-x^2,\]

inside which an isosceles triangle \(OAB\) has been drawn.

Triangle OAB inside y equals 15 minus x squared The isosceles triangle has vertex O at the origin and the other vertices A and B at equal heights on opposite sides of the parabola. A B O x y y = 15 − x²

Find the maximum possible area, \(A\), of the triangle. You may assume that your answer is a maximum.

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

First walkthrough idea

Focus to try first

Express the triangle's base and height using \(x\), build one area function, and maximise it.

Step 1

Build the area function

The base \(AB\) is \(2x\), and the height from \(O\) is \(y=15-x^2\).

Show the first step’s working
\[A=\frac12(2x)y=xy\] \[A=x(15-x^2)=15x-x^3\]

Walkthrough overview

What this question practises

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

Method: Maximising the area of a triangle inside a parabola.

This is Question 3(c) 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 maximising the area of a triangle inside a parabola. 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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