Mathematics • Year 9 • Unit 3 • Lesson 9
Tangent in the Real World — Towers, Cliffs and Slopes
Apply tan θ = opp/adj to building heights from a ground distance, road-grade calculations, lighthouse height-finding, surveying a tree, and the world's most photographed structure — the Sydney Harbour Bridge approach grade.
1. Word problems
Calculator in DEG mode. Round to 2 d.p. unless told otherwise.
1.1 — Sydney Tower from Circular Quay. Standing on the Circular Quay foreshore 350 m horizontally from the base of Sydney Tower, you look up at the top observation deck at an angle of elevation of 48°.
(a) Find the height of the observation deck above eye level.
(b) Your eye is 1.6 m above the ground. Find the total height of the deck above the ground. 3 marks
1.2 — Driveway grade. A new driveway will drop 1.8 m vertically from the garage to the street. The driveway will be straight, and the council's maximum allowed driveway slope is 1:8 (i.e. 1 m vertical drop for every 8 m horizontal run), which corresponds to about θ = 7.13°.
(a) Find the minimum horizontal run for the driveway, using tan 7.13° ≈ 0.125.
(b) If the architect wants a gentler 1:12 grade (about θ = 4.76°, tan ≈ 0.0833) instead, find the new horizontal run. 3 marks
1.3 — Cape Byron lighthouse. A boat is exactly 800 m from the base of Cape Byron Lighthouse (at sea level). The skipper sees the top of the lighthouse at an angle of elevation of 8°.
(a) Find the height of the lighthouse above sea level.
(b) The boat moves to a point 600 m from the base. What's the new angle of elevation? (Use tan⁻¹.) 3 marks
1.4 — School oval gum tree. A botanist wants to estimate the height of a 100-year-old gum tree on the school oval without climbing it. She measures 18 m from the base, holds her clinometer at eye level (1.5 m above the ground) and reads an angle of 52°.
(a) Find the height of the tree above her eye level.
(b) Find the total height of the tree from the ground. 3 marks
1.5 — Harbour Bridge approach grade. The Sydney Harbour Bridge's southern road approach rises from Dawes Point at an average grade of 3.3° over a horizontal distance of approximately 290 m to the bridge deck.
(a) Find the vertical rise from Dawes Point to the bridge deck.
(b) If a cyclist wants to know how steep they'll have to ride, would they describe 3.3° as "gentle", "moderate" or "steep"? Justify in one sentence by comparing to a 1:14 (4°) wheelchair ramp. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 Your classmate calculates a building height as 25 × tan 35° ≈ 17.5 m, then writes "Total building height = 17.5 m" — without adding their eye height. The teacher marks the answer wrong. In your own words, explain (i) why the answer is incomplete, (ii) what part of the building the calculated 17.5 m actually represents, and (iii) how a real surveyor would correct it. Refer to "above eye level" and "eye height" somewhere in your explanation.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Sydney Tower from Circular Quay
(a) Above eye level: 350 tan 48° ≈ 350 × 1.1106 ≈ 388.71 m.
(b) Total above ground: 388.71 + 1.6 ≈ 390.31 m.
Sydney Tower is actually about 305 m — the 48° angle assumed here is illustrative, not measured.
1.2 — Driveway grade
(a) 1:8 grade: horizontal run = 1.8 / tan 7.13° ≈ 1.8 / 0.125 = 14.40 m.
(b) 1:12 grade: horizontal run = 1.8 / tan 4.76° ≈ 1.8 / 0.0833 ≈ 21.61 m.
The gentler grade needs about 7 extra metres of run — important if the property is small.
1.3 — Cape Byron Lighthouse
(a) Height = 800 tan 8° ≈ 800 × 0.1405 ≈ 112.43 m above sea level.
(b) tan θ_new = 112.43 / 600 ≈ 0.1874. θ_new = tan⁻¹(0.1874) ≈ 11°.
Closer boat → steeper look-up angle, as expected.
1.4 — Gum tree
(a) Above eye level: 18 tan 52° ≈ 18 × 1.2799 ≈ 23.04 m.
(b) Total tree height: 23.04 + 1.5 = 24.54 m.
Mature blue gums (Eucalyptus globulus) often reach this height — entirely realistic.
1.5 — Harbour Bridge approach
(a) Vertical rise = 290 tan 3.3° ≈ 290 × 0.0577 ≈ 16.72 m.
(b) A 3.3° grade is gentle — it's actually slightly less steep than the 4° (1:14) wheelchair ramp maximum. Cyclists describe it as "easy on the bridge" — the long approach makes it feel imperceptible.
2.1 — Explain your thinking (sample response)
The classmate's answer is incomplete because the calculation 25 × tan 35° only gives the part of the building above eye level, not the part from the ground up to eye level. The horizontal distance (adj = 25 m) was measured from the observer's feet, but the angle of elevation was measured from the observer's eyes — so the right triangle the calculation describes has its bottom edge at eye-height, not on the ground. To get the full building height from the ground, you have to add the observer's eye height (typically 1.5–1.7 m) to the calculated number, giving a total of about 17.5 + 1.6 ≈ 19.1 m. A real surveyor would either record the eye height alongside each measurement and add it at the end, or use a tripod-mounted instrument whose height above the ground is already known.
Marking: 1 mark for naming the missing eye height; 1 mark for saying calculated value is "above eye level"; 1 mark for the corrected total = (calculated) + (eye height); 1 mark for clear full-sentence explanation.