Level 3 Differentiation Walkthrough

2021 NCEA Level 3 Differentiation Question 3(e)

2021 Paper

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Question

A lamp is suspended above the centre of a round table of radius \(r\). The height, \(h\), of the lamp above the table is adjustable.

light source h r S θ P

Point \(P\) is on the edge of the table.

At point \(P\), the illumination \(I\) is directly proportional to the cosine of angle \(\theta\) in the above diagram, and inversely proportional to the square of the distance, \(S\), to the lamp.

\[ I=\frac{k\cos\theta}{S^2}, \]

where \(k\) is a constant.

Prove that the edge of the table will have maximum illumination when \(h=\frac{r}{\sqrt{2}}\).

You do not need to prove that your solution gives the maximum value. You must use calculus and show any derivatives that you need to find when solving this problem.

First walkthrough idea

Focus to try first

Rewrite the illumination formula using only \(h\) and the fixed table radius \(r\), then differentiate with respect to \(h\).

Step 1

Rewrite the geometry

Use the right triangle in the diagram.

Show the first step’s working
\[ \cos\theta=\frac{h}{S} \] \[ S^2=h^2+r^2 \] \[ S=(h^2+r^2)^{1/2} \]

Walkthrough overview

What this question practises

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

Method: Lamp and table optimisation using product and chain rules.

This is Question 3(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.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise lamp and table optimisation using product and chain rules. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Do not stop after differentiating the outside function; include the derivative of the inside function as a factor.

Continue practising

← Question 3(d)