Level 3 Differentiation Walkthrough

2023 NCEA Level 3 Differentiation Question 3(c)

2023 Paper

← Back to paper

Question

Char goes for a ride on a Ferris wheel. As she rotates around, her position can be described by the pair of parametric equations

\[ x=5\sqrt{2}\sin\left(\frac{\pi t}{5}\right) \] \[ y=10-5\sqrt{2}\cos\left(\frac{\pi t}{5}\right) \]

where \(t\) is time, in seconds, from the start of the ride.

Find the gradient of the normal to this curve at the point when \(t=6.25\) seconds.

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

First walkthrough idea

Hint to try first

Differentiate \(x\) and \(y\) with respect to \(t\) first.

Step 1

Find the tangent gradient

The common factor of \(\pi\sqrt{2}\) cancels out.

Show the first step’s working
\[ \frac{dx}{dt}=\pi\sqrt{2}\cos\left(\frac{\pi t}{5}\right) \qquad \frac{dy}{dt}=\pi\sqrt{2}\sin\left(\frac{\pi t}{5}\right) \]

The common factor of \(\pi\sqrt{2}\) cancels out.

Key result

\[ \frac{dy}{dx}=\tan\left(\frac{\pi t}{5}\right) \]

Walkthrough overview

What this question practises

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

Method: Parametric differentiation for a Ferris wheel model.

This is Question 3(c) 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.

Page updated .

Learning summary

Review the method, not only the answer

This walkthrough helps you practise parametric differentiation for a Ferris wheel model. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 3(b) Question 3(d) →