Case Study – 1. An injured bird was found on the roof of a building. The building is 15 m high. A fireman was called to rescue the bird. The fireman used an adjustable ladder to reach the roof. He placed the ladder in such a way that the ladder makes an angle of 60° with the ground in order to reach the roof. Based on the above information, answer the following questions : (i) Find the length of the ladder used by the fireman to reach the roof. (ii) Find the distance of the point on the ground at which the ladder was fixed from the bottom of the building. (iii)(a) In order to avoid skidding, the fireman placed the ladder in such a way that the bottom of the ladder touches the base of the wall which is opposite to the building, making an angle of 30° with the ground. Draw a neat diagram to represent the above situation and hence find the width of the road between the building and the wall. (iii)(b) In order to avoid skidding, the fireman placed the ladder in such a way that the bottom of the ladder touches the base of the wall which is opposite to the building, making an angle of 30° with the ground. Find the length of the ladder used by the fireman in this case.

MathsCBSE Class 10 · Basic · Board exam 2025, set 430/1/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Basic 2025: question 36

Set in the CBSE Class 10 board exam of 2025, paper 430/1/1, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedSome Applications of Trigonometry

Case Study – 1. An injured bird was found on the roof of a building. The building is 15 m high. A fireman was called to rescue the bird. The fireman used an adjustable ladder to reach the roof. He placed the ladder in such a way that the ladder makes an angle of $6 0^{\circ}$ with the ground in order to reach the roof. Based on the above information, answer the following questions :

(i)
Find the length of the ladder used by the fireman to reach the roof.
1 mark
(ii)
Find the distance of the point on the ground at which the ladder was fixed from the bottom of the building.
1 mark
(iii)(a)
In order to avoid skidding, the fireman placed the ladder in such a way that the bottom of the ladder touches the base of the wall which is opposite to the building, making an angle of $3 0^{\circ}$ with the ground. Draw a neat diagram to represent the above situation and hence find the width of the road between the building and the wall.
2 marks
OR
(iii)(b)
In order to avoid skidding, the fireman placed the ladder in such a way that the bottom of the ladder touches the base of the wall which is opposite to the building, making an angle of $3 0^{\circ}$ with the ground. Find the length of the ladder used by the fireman in this case.
2 marks

Figure in the paper: Photograph of a bird on a roof and a fireman on a ladder; a sketch shows a ladder leaning against a vertical wall, with its foot on the ground and making an angle of 60° with the ground. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 430/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Draw a right triangle: the building is the vertical side (15 m), the ground is the base, and the ladder is the hypotenuse.

Hint 2

For the ladder (the hypotenuse), use sine. For the distance along the ground, use tangent.

Hint 3

In part (iii), use the same triangle but with a $3 0^{\circ}$ angle at the foot of the ladder.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

(i) Let AB be the building of height 15 m and AC the ladder of length $a$ , with $∠ A C B = 6 0^{\circ}$ and $∠ B = 9 0^{\circ}$ . In right $△ A B C$ :
$sin 6 0^{\circ} = \frac{A B}{A C} \Rightarrow \frac{3}{2} = \frac{15}{a}$
½ mark
(i) $a = \frac{30}{3} = 103$
So the length of the ladder is $103$ m (about 17.32 m).
½ mark
(ii) Let BC $= x$ be the distance of the foot of the ladder from the building. Then
$tan 6 0^{\circ} = \frac{A B}{B C} \Rightarrow 3 = \frac{15}{x}$
½ mark
(ii) $x = \frac{15}{3} = 53$
So the foot of the ladder is $53$ m (about 8.66 m) from the bottom of the building.
½ mark
(iii)(a) Diagram: Draw a vertical segment AB = 15 m for the building, with B on the ground. Draw the horizontal road BC, where C is the base of the opposite wall, and draw a short vertical segment at C for the wall. Join the ladder AC and mark $∠ A C B = 3 0^{\circ}$ and $∠ A B C = 9 0^{\circ}$ . Let BC $= y$ be the width of the road.
1 mark
(iii)(a) In right $△ A B C$ :
$tan 3 0^{\circ} = \frac{A B}{B C} \Rightarrow \frac{1}{3} = \frac{15}{y}$
½ mark
(iii)(a) $y = 153$
So the width of the road is $153$ m (about 25.98 m).
½ mark
(iii)(b) Use the same right $△ A B C$ with AB = 15 m, $∠ A C B = 3 0^{\circ}$ and ladder AC $= l$ :
$sin 3 0^{\circ} = \frac{A B}{A C} \Rightarrow \frac{1}{2} = \frac{15}{l}$
1 mark
(iii)(b) $l = 30$
So the length of the ladder in this case is 30 m.
1 mark

Answer: (i) $103$ m; (ii) $53$ m; (iii)(a) $153$ m; (iii)(b) 30 m

Where the marks are

(i) Uses $sin 6 0^{\circ} = 15/ a$ (½ mark)
(i) Gets ladder length $103$ m (½ mark)
(ii) Uses $tan 6 0^{\circ} = 15/ x$ (½ mark)
(ii) Gets distance $53$ m (½ mark)
(iii)(a) Draws a correct labelled diagram (1 mark)
(iii)(a) Uses $tan 3 0^{\circ} = 15/ y$ (½ mark)
(iii)(a) Gets road width $153$ m (½ mark)
(iii)(b) Uses $sin 3 0^{\circ} = 15/ l$ (1 mark)
(iii)(b) Gets ladder length 30 m (1 mark)

Revise the chapter

Chapter: Some Applications of Trigonometry (notes, formulas, where marks are lost and every past question)
Topic: Line of Sight and Angle of Elevation

Practise this chapterBoard-style questions from Some Applications of Trigonometry, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 430/1/1 (Basic): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Basic 2025: question 36

Hints

Step-by-step solution

Revise the chapter

More questions from Some Applications of Trigonometry