Level 3 Differentiation Walkthrough

2017 NCEA Level 3 Differentiation Question 3(d)

2017 Paper

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Question

A building has an external elevator. The elevator is rising at a constant rate of \(2\text{ m s}^{-1}\). Sarah is stationary, watching the elevator from a point \(30\text{ m}\) away from the base of the elevator shaft.

Let the angle of elevation of the elevator floor from Sarah's eye level be \(\theta\).

Elevator, Sarah, and angle of elevation Sarah stands thirty metres horizontally from an elevator shaft. The elevator floor is x metres above her eye level and rises at two metres per second. A dashed line of sight forms the angle theta at Sarah. rising at 2 m s⁻¹ x 20 m at this instant 30 m Sarah θ elevator shaft elevator floor

Find the rate at which the angle of elevation is increasing when the elevator floor is \(20\text{ m}\) above Sarah's eye level.

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

First walkthrough idea

Focus to try first

Let the vertical height \(x\) and angle \(\theta\) depend on time. Connect them with tangent, then differentiate with respect to time.

Step 1

Define the changing quantities

The height and viewing angle change with time; the horizontal distance stays fixed.

Show the first step’s working

Let \(x=x(t)\) be the elevator floor's height above Sarah's eye level and let \(\theta=\theta(t)\).

\[\frac{dx}{dt}=2\text{ m s}^{-1},\qquad 30\text{ m is constant}.\]

The requested rate is \(d\theta/dt\), not the elevator's vertical speed \(dx/dt\).

Walkthrough overview

What this question practises

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

Method: Related rates for an elevator and a changing viewing angle.

This is Question 3(d) 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 related rates for an elevator and a changing viewing angle. 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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