Level 3 Differentiation Walkthrough

2018 NCEA Level 3 Differentiation Question 1(d)

2018 Paper

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Question

Car, rope, and pulley A rope of length L runs from the tow-bar of a car to a pulley three metres higher. The horizontal separation is x. L x 3 m tow-bar pulley

A car is being pulled along by a rope attached to the tow-bar at the back of the car. The rope passes through a pulley, the top of which is \(3\text{ m}\) further from the ground than the tow-bar.

The pulley is \(x\text{ m}\) horizontally from the tow-bar. The rope is being winched in at a speed of \(0.6\text{ m s}^{-1}\), and the wheels remain in contact with the ground.

At what speed is the car moving when the length \(L\) between the tow-bar and pulley is \(5.4\text{ m}\)?

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

First walkthrough idea

Focus to try first

The rope, ground, and fixed \(3\text{ m}\) height form a right triangle. Relate \(L\) and \(x\), then relate their rates.

Step 1

Write the geometric relationship

Apply Pythagoras to the right triangle.

Show the first step’s working
\[L^2=x^2+3^2=x^2+9\]

Walkthrough overview

What this question practises

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

Method: Related rates for a car, rope, and pulley.

This is Question 1(d) 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 related rates for a car, rope, and pulley. 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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