Passenger boarding stairs, sometimes referred to as boarding ramps, stair cars or aircraft steps, provide a mobile means to travel between the aircraft doors and the ground. Larger aircraft have door sills 5 to 20 feet (1 foot = 30 cm) high. Stairs facilitate safe boarding and de-boarding. An aircraft has a door sill at a height of 15 feet above the ground. A stair car is placed at a horizontal distance of 15 feet from the plane.
Based on given information, answer the questions given in part (i) and (ii).
Generated by claude-sonnet-4-6 · 2026-06-15 10:37 · grounding stimulus
Model Answer
(i) Let the angle of inclination be θ.
Given: height = 15 ft, horizontal distance = 15 ft
$$\tan\theta = \frac{15}{15} = 1 \Rightarrow \theta = 45°$$
The stairs are inclined at 45°.
---
(ii) Using Pythagoras theorem:
$$\text{Length of stairs} = \sqrt{15^2 + 15^2} = \sqrt{225 + 225} = \sqrt{450} = 15\sqrt{2} \text{ feet}$$
The length of stairs is $15\sqrt{2}$ feet.
---
(iii) Given: length of stairs = 20 ft, angle = 60°
$$\sin 60° = \frac{\text{height}}{20}$$
$$\frac{\sqrt{3}}{2} = \frac{h}{20}$$
$$h = 10\sqrt{3} = 10 × 1.732 = \textbf{17.32 feet}$$
The height of the door sill is 17.32 feet.
Source: Applications of Trigonometry, Chapter 9
---
Explanation
- Part (i): Since height = base = 15 ft, tan θ = 1, so θ = 45°. This is a standard trigonometric value.
- Part (ii): The stairs form the hypotenuse; use Pythagoras or note that for a 45° right triangle, hypotenuse = side × √2.
- Part (iii): The stairs are the hypotenuse (20 ft). Use sin (opposite/hypotenuse) since height is opposite the given angle. Substitute √3 = 1.732 as instructed. Examiners award 1 mark for correct formula and 1 mark for final answer.