Mathematics • Year 10 • Unit 3 • Lesson 4

Elevation & Depression in the Real World

Apply Lesson 4 to real Australian situations — sighting a fire from a Snowy Mountains lookout, the angle a Sydney Tower observation-deck visitor sees, the rise of a hot-air balloon at the Canberra Balloon Spectacular. Each problem requires a labelled diagram with the horizontal marked, then the right trig ratio. Round to 1 d.p. for lengths and to the nearest degree for angles.

Apply · Real-World Maths

1. Word problems

For each problem: (i) sketch the situation, marking the horizontal and the line of sight, (ii) write the angle of elevation or depression on the diagram, (iii) identify the right-angled triangle, (iv) choose sin, cos or tan, (v) calculate. Diagrams are worth marks.

1.1 — Sydney Tower visitor. A visitor on the Sydney Tower Eye observation deck is 251 m above street level. She looks down at the Queen Victoria Building entrance and the angle of depression is 35°. Find the horizontal ground distance from the foot of the Sydney Tower directly below the visitor to the QVB entrance. Round to the nearest metre. 3 marks

Stuck? Tower height = opposite = 251. Distance = adjacent. tan 35° = 251 / distance (using alternate-angle reasoning).

1.2 — Hot-air balloon. At the Canberra Balloon Spectacular, an observer 120 m horizontally from a point directly below a balloon measures the angle of elevation to the basket as 22°. Find the height of the balloon above the ground, to 1 d.p. 3 marks

Stuck? Horizontal distance = adjacent. Balloon height = opposite. tan 22° = h / 120.

1.3 — Fire tower in the Snowy Mountains. A fire-spotter in a 25 m tower spots smoke. The angle of depression from her observation window (at the top of the tower) to the smoke is 8°. Assuming the ground is level, find the ground distance from the foot of the tower to the smoke. Round to the nearest metre. 3 marks

Stuck? Use the alternate-angles trick. The angle of elevation at the smoke is also 8°. tan 8° = 25 / distance.

1.4 — Surf-life-saving observation. A lifeguard on a 4 m observation chair at Bondi spots a swimmer 95 m horizontally from the chair. Find the angle of depression from the lifeguard's eye level (at the top of the chair) to the swimmer, to the nearest degree. 3 marks

Stuck? tan θ = 4 / 95. Use tan⁻¹ to find θ.

1.5 — Two-step: approaching a building. A pedestrian walks toward an office tower along a level footpath. At a first point, the angle of elevation to the top of the tower is 20°. After walking 50 m closer (along the same footpath), the angle of elevation becomes 40°. Find the height of the tower. (Hint: set up two tan equations with shared unknown distance, then equate the heights.) 4 marks

Stuck? Let x be the distance from the closer point to the base. Then h = x × tan 40° and h = (x + 50) × tan 20°.

2. Explain your thinking

This question is about communication, not just numbers. Use full sentences. 4 marks

2.1 A classmate says: "The angle of depression from the top of a cliff to a boat is 30°. So the angle inside the right-angled triangle (at the top of the cliff, between the cliff and the line of sight) is also 30°." Explain in 4-6 sentences why this is wrong, using the definitions in Lesson 4. Refer to where the angle of depression is measured from (which line) and to the alternate-angles property used to transfer the angle into the triangle. State the correct angle inside the triangle at the top of the cliff.

Stuck? Angle of depression is measured downward from the horizontal, not from the vertical cliff. The triangle has the angle of depression outside it (between the horizontal and the line of sight), and the angle at the top between the cliff and the line of sight is the complement: 90° − 30° = 60°.

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Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Sydney Tower to QVB

Angle inside triangle at QVB = 35° (alternate angles).
tan 35° = 251 / distance → distance = 251 / tan 35° = 251 / 0.7002 = 358.46 ≈ 358 m.

1.2 — Balloon height

tan 22° = h / 120 → h = 120 × tan 22° = 120 × 0.4040 = 48.49 ≈ 48.5 m.

1.3 — Fire-spotter

tan 8° = 25 / distance → distance = 25 / tan 8° = 25 / 0.1405 = 177.86 ≈ 178 m.
A shallow depression angle (8°) means the smoke is well away.

1.4 — Bondi lifeguard angle

tan θ = 4 / 95 = 0.04211, θ = tan⁻¹(0.04211) = 2.41° ≈ 2°.
Very small depression — the lifeguard's chair is only 4 m up, the swimmer is far out.

1.5 — Two-step pedestrian

Let x = distance from closer point to base.
From closer point: h = x × tan 40° = x × 0.8391.
From further point: h = (x + 50) × tan 20° = (x + 50) × 0.3640.
Set equal: 0.8391 x = 0.3640 x + 18.20
0.4751 x = 18.20 → x = 38.31 m.
Height h = 38.31 × tan 40° = 38.31 × 0.8391 = 32.14 ≈ 32.1 m.
Check: from 88.31 m, tan 20° × 88.31 = 0.3640 × 88.31 ≈ 32.14 m. ✓

2.1 — Cliff classmate (sample response)

The classmate is wrong. The angle of depression is measured from the horizontal at the observer downward to the line of sight — not from the vertical cliff. Inside the right-angled triangle (with the cliff as one side, the line of sight as the hypotenuse, and the ground as the third side), the 30° depression sits outside the triangle, between the horizontal at the top of the cliff and the line of sight. To get the angle inside the triangle at the top of the cliff (between the cliff and the line of sight), use the fact that the cliff is vertical and the horizontal is at 90° to it: so the inside angle is 90° − 30° = 60°. (We could also use alternate angles to transfer the 30° down to the boat, where it becomes the angle of elevation at the boat — also 30°.)

Marking: 1 for naming the horizontal as the reference, 1 for explaining the angle is outside the triangle, 1 for the 60° inside the triangle, 1 for the alternate-angles transfer.