Level 3 Differentiation Walkthrough

2016 NCEA Level 3 Differentiation Question 3(e)

2016 Paper

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Question

In a rugby game, a try is scored \(15\text{ m}\) from the left-hand goal-post. The conversion kick is taken at some point on the line perpendicular to the goal-line from the point where the try was scored, as shown in the diagram below.

The ball needs to pass between the goal-posts, which are \(5.4\text{ m}\) apart.

Rugby conversion angle geometry The ball is on a line perpendicular to the goal-line through the point where the try was scored. That point is fifteen metres from the nearer goal-post, and the posts are five point four metres apart. The ball is d metres from the goal-line, and the two sight lines from the ball to the posts form angle theta. goal-line α θ d 15 m 5.4 m ball

Find the distance \(d\) from the goal-line that the conversion kick should be taken from in order to maximise the angle \(\theta\) between the lines from the ball to the goal-posts.

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

You do not need to prove that the angle you have found is a maximum.

First walkthrough idea

Focus to try first

Express the viewing angle through tangent subtraction, then maximise the resulting one-variable function of d.

Step 1

Write the two sight-line angles

Use the right triangles made by the perpendicular distance d and the distances along the goal-line.

Show the first step’s working
\[\tan\alpha=\frac{15}{d}\] \[\tan(\alpha+\theta)=\frac{20.4}{d}.\]

Walkthrough overview

What this question practises

This 2016 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Maximising a rugby conversion angle using trigonometry and calculus.

This is Question 3(e) from the 2016 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 maximising a rugby conversion angle using trigonometry and calculus. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Finding a stationary value is only part of an optimisation argument; justify that it is the required maximum and respect the domain.

Continue practising

← Question 3(d)