The angles of depression of the top and the bottom of an 8 m tall building from the top of another multistoried building are 30° and 45°, respectively. Find the height of the multistoried building and the distance between the two buildings.

MathsCBSE Class 10 · Standard · Board exam 2025, set 30/2/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2025: question 32(b)

Set in the CBSE Class 10 board exam of 2025, paper 30/2/1, section D, question 32(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(b) · 5 marks · Long answerCompetency-basedSome Applications of Trigonometry

The angles of depression of the top and the bottom of an 8 m tall building from the top of another multistoried building are $3 0^{\circ}$ and $4 5^{\circ}$ , respectively. Find the height of the multistoried building and the distance between the two buildings.

Figure in the paper: A tall multistoried building on the left and a shorter 8 m building on the right on the same ground. From the top of the taller building, horizontal line of sight; angles of depression $3 0^{\circ}$ to the top of the 8 m building and $4 5^{\circ}$ to its bottom are marked. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/2/1.

Hints

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

Hint 1

Draw a horizontal line from the top of the 8 m building to the tall building; this forms a rectangle.

Hint 2

In the upper triangle the vertical side is (h − 8); in the lower triangle it is h. Both share the same horizontal distance.

Hint 3

Use tan 30° and tan 45°, equate the two expressions for the horizontal distance, then solve and rationalise.

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.

Figure: Let CD be the multistoried building, with top D and foot C. Let AB be the 8 m building, with top A and foot B. Draw AE ∥ BC meeting CD at E, so ABCE is a rectangle with $E C = A B = 8$ m and $A E = B C$ .
The angles of depression from D are $3 0^{\circ}$ to A and $4 5^{\circ}$ to B. By alternate angles, $∠ D A E = 3 0^{\circ}$ and $∠ D B C = 4 5^{\circ}$ .
Let $D C = h$ m. Then $D E = h - 8$ .
1 mark
In right $△ D E A$ :
$\frac{D E}{A E} = tan 3 0^{\circ} = \frac{1}{3}$
1 mark
$\Rightarrow A E = 3 D E$ ... (1)
½ mark
In right $△ D C B$ :
$\frac{D C}{B C} = tan 4 5^{\circ} = 1$
½ mark
$\Rightarrow B C = D C$ ... (2)
½ mark
Since $A E = B C$ :
$3 (D C - 8) = D C \Rightarrow D C (3 - 1) = 83 \Rightarrow D C = \frac{8 3}{3 - 1}$
½ mark
Rationalising:
$D C = \frac{8 3 ( 3 + 1 )}{( 3 - 1 ) ( 3 + 1 )} = \frac{8 ( 3 + 3 )}{2} = (12 + 43) m \approx 18.93 m$
½ mark
From (2), the distance between the buildings is $B C = D C = (12 + 43)$ m $\approx 18.93$ m.
½ mark

Answer: Height of the multistoried building = $(12 + 43)$ m ≈ 18.93 m; distance between the buildings = $(12 + 43)$ m ≈ 18.93 m

Where the marks are

Correct labelled figure (1 mark)
Uses $tan 3 0^{\circ} = \frac{D E}{A E}$ (1 mark)
Gets $A E = 3 D E$ (1/2 mark)
Uses $tan 4 5^{\circ} = \frac{D C}{B C}$ (1/2 mark)
Gets $B C = D C$ (1/2 mark)
Forms $3 (D C - 8) = D C$ and gets $D C = \frac{8 3}{3 - 1}$ (1/2 mark)
Rationalises to $12 + 43$ m (1/2 mark)
Distance $B C = (12 + 43)$ m (1/2 mark)

Revise the chapter

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

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 30/2/1 (Standard): 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 Standard 2025: question 32(b)

Hints

Step-by-step solution

Revise the chapter

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