Level 3 Differentiation Walkthrough

2023 NCEA Level 3 Differentiation Question 2(e)

2023 Paper

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Question

A police helicopter is flying above a straight horizontal section of motorway chasing a speeding car.

The helicopter is flying at a constant speed of \(72\text{ m s}^{-1}\) and at a constant height of \(400\) metres above the ground.

When the direct distance from the helicopter to the car is \(2500\) metres, the angle of depression \(\theta\) between the horizontal and the line of sight from the helicopter to the car is increasing at a rate of \(0.002\text{ rad s}^{-1}\).

Calculate the speed of the car at this instant.

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

First walkthrough idea

Hint to try first

Let \(x\) be the horizontal distance from the helicopter to the car, so \(\tan\theta=\frac{400}{x}\).

Step 1

Link the angle and the horizontal distance

The opposite side is the constant height \(400\), and the adjacent side is the horizontal distance \(x\).

Show the first step’s working

The opposite side is the constant height \(400\), and the adjacent side is the horizontal distance \(x\).

Key result

\[ \tan\theta=\frac{400}{x} \]

Walkthrough overview

What this question practises

This 2023 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Related rates for a helicopter chasing a car.

This is Question 2(e) from the 2023 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 related rates for a helicopter chasing a car. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Differentiate with respect to time consistently, then include the correct units and contextual interpretation.

Continue practising

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