Mathematics • Year 10 • Unit 3 • Lesson 19
Surveying, Navigation, Heights — Real Australian Problems
Apply Lesson 19 to authentic situations — a Pittwater sailing course, a Centrepoint Tower elevation, a bushwalking three-leg traverse, an outback windmill height, and a Bondi cliff lookout. Draw the diagram, label everything (North, distances, bearings), pick the right rule, then round to a sensible number of significant figures.
1. Word problems
For each problem: (i) draw the diagram with all distances, bearings and angles labelled, (ii) identify the pattern (right-angled trig / sine rule / cosine rule), (iii) calculate, (iv) round sensibly.
1.1 — Pittwater sailing. A yacht leaves Newport on bearing 080° for 30 km, then turns and sails on bearing 150° for 40 km.
(a) Find the interior angle of the triangle at the turn.
(b) Find the direct distance back to Newport.
(c) Find the return bearing the skipper should steer. 5 marks
1.2 — Centrepoint Tower elevation. Two students standing 60 m apart on Castlereagh Street measure the angle of elevation to the top of Sydney's Centrepoint Tower. The closer student measures 78°, the farther student (60 m further back) measures 76°.
(a) Set up two equations for the tower height h.
(b) Solve for h to the nearest metre. 4 marks
1.3 — Royal National Park bushwalk. A bushwalker walks 4 km on bearing 070°, then 3 km on bearing 160°. (The change in bearing is 90°, so the interior angle at the turn is 90°.)
(a) How far is she from the start, in a straight line?
(b) What bearing should she walk to return to the start? 4 marks
1.4 — Outback windmill height. A windmill on a Western NSW property is being surveyed. From a point 50 m from the base on flat ground, the angle of elevation to the top of the windmill blades is 22°, and to the top of the supporting tower (just below the blades) is 18°.
(a) Find the height to the top of the blades.
(b) Find the height to the top of the tower.
(c) Find the length of the blade-and-hub assembly (the difference). 3 marks
1.5 — Bondi cliff lookout. From a Bondi clifftop 50 m above sea level, the angle of depression to a yacht is 12°.
(a) How far is the yacht from the base of the cliff (horizontally)?
(b) Five minutes later, the angle of depression to the same yacht is 28°. How far has the yacht sailed toward the cliff in those five minutes? 3 marks
2. Explain your thinking
This question is about communication and precision. Use full sentences. 4 marks
2.1 A junior surveyor reports a property boundary length as "4 217.83 m", based on a tape measure accurate to the nearest metre. In 4-6 sentences, explain (i) why this reported number is misleading, (ii) what number of significant figures would be honest for that measurement, (iii) the difference between a bearing and an interior angle in a triangle (and why students often confuse them), and (iv) one practical step a surveyor should take before quoting any final answer. Use the words "significant figures" and "back-bearing" in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Pittwater yacht (30 km on 080°, 40 km on 150°)
(a) Interior angle at the turn = 180° − (150° − 80°) = 110°.
(b) d² = 30² + 40² − 2(30)(40) cos 110° = 900 + 1600 − 2400 × (−0.342) = 2500 + 820.8 = 3320.8. d ≈ 57.6 km.
(c) Sine rule for the angle at the start: sin θ / 40 = sin 110° / 57.6. sin θ = 40 × 0.9397 / 57.6 ≈ 0.6526. θ ≈ 40.7°. Bearing from start to finish = 080° + 40.7° = 120.7°. Return bearing (back-bearing) = 120.7° + 180° = 300.7° ≈ 301°.
1.2 — Centrepoint Tower (elevations 78° and 76°, baseline 60 m)
(a) tan 78° = h / x → h = x × 4.7046; tan 76° = h / (x + 60) → h = (x + 60) × 4.0108.
(b) Equate: 4.7046 x = 4.0108 (x + 60) = 4.0108 x + 240.65.
0.6938 x = 240.65 → x ≈ 346.9 m. h = 346.9 × 4.7046 ≈ 1 632 m.
(Note: this is much taller than the real Centrepoint Tower — the small angle difference makes the system very sensitive. In practice you would survey from a longer baseline. Treat this answer as the mathematical solution to the given numbers.)
1.3 — Royal National Park (4 km on 070°, 3 km on 160°)
(a) Interior angle at turn = 90°. d = √(16 + 9) = √25 = 5 km.
(b) Angle off the first leg back to start: tan α = 3/4 → α = 36.9°. Bearing of start from finish: facing along reverse of leg 2 (340°), rotate further by 36.9° toward leg 1, giving… easier method: bearing finish→start = (bearing start→finish) + 180°. Bearing start→finish = 070° + 36.9° = 106.9°. Back-bearing = 106.9° + 180° = 286.9° ≈ 287°.
1.4 — Windmill (50 m horizontal, elevations 22° and 18°)
(a) Top of blades: h₁ = 50 × tan 22° = 50 × 0.4040 ≈ 20.2 m.
(b) Top of tower: h₂ = 50 × tan 18° = 50 × 0.3249 ≈ 16.2 m.
(c) Blade-and-hub assembly ≈ 20.2 − 16.2 = 4.0 m.
1.5 — Bondi cliff (50 m above water, depressions 12° then 28°)
(a) d₁ = 50 / tan 12° = 50 / 0.2126 ≈ 235.2 m.
(b) d₂ = 50 / tan 28° = 50 / 0.5317 ≈ 94.0 m. Distance sailed = 235.2 − 94.0 ≈ 141 m.
2.1 — Explain your thinking (sample response)
Reporting "4 217.83 m" is misleading because the tape measure is only accurate to the nearest metre — the digits to the right of the decimal point are noise, not measurements. An honest answer would be to 3 or 4 significant figures, i.e. "4 220 m" or "4 217 m", matching the precision of the underlying data. A bearing is always measured clockwise from North (000° to 360°), so the same physical line has two different bearings depending on which end you stand at (the second bearing being the back-bearing, ±180° from the first). An interior angle of a triangle, by contrast, is the rotation between two path legs inside the triangle; it is not a compass direction. Students often confuse the two by treating a bearing as if it were an interior angle. Before quoting any final answer, a surveyor should at minimum (a) sketch a labelled diagram, (b) state the round-off rule that matches the input precision (e.g. nearest metre), and (c) verify with an alternative method such as an East/North components check.
Marking: 1 for spotting the false precision, 1 for naming an appropriate sig-fig answer, 1 for clarifying bearing vs interior angle (with back-bearing reference), 1 for a sensible verification step.