MathsCBSE Class 10 · Basic · Board exam 2024, set 430/3/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2024: question 34

Set in the CBSE Class 10 board exam of 2024, paper 430/3/1, section D, question 34, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q34 · 5 marks · Long answerCompetency-basedSome Applications of Trigonometry

Two poles of equal height are standing opposite each other on either side of a road, which is $100$ m wide. From a point somewhere between them on the road, the angles of elevation of the top of the poles are $6 0^{\circ}$ and $3 0^{\circ}$ respectively. Find the height of the poles and the distances of the point from the poles.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/3/1.

Hints

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

Hint 1

Draw both poles and the point on the road between them; you get two right triangles with the same height.

Hint 2

Write $tan 6 0^{\circ}$ and $tan 3 0^{\circ}$ in terms of $h$ and the two unknown distances.

Hint 3

The two distances add up to the width of the road, 100 m.

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 AB and CD be the poles, each of height $h$ m, standing at A and C on opposite sides of the road, $A C = 100$ m. P is a point on AC with $P A = y$ m and $P C = x$ m, $∠ B P A = 6 0^{\circ}$ , $∠ D P C = 3 0^{\circ}$ , and $∠ A = ∠ C = 9 0^{\circ}$ .
1 mark
In right $△ B A P$ : $tan 6 0^{\circ} = \frac{h}{y} \Rightarrow h = 3 y$ ...(i)
1 mark
In right $△ D C P$ : $tan 3 0^{\circ} = \frac{h}{x} \Rightarrow h = \frac{x}{3}$ ...(ii)
1 mark
From (i) and (ii): $3 y = \frac{x}{3} \Rightarrow x = 3 y$ . Also $x + y = 100$ .
1 mark
$3 y + y = 100 \Rightarrow y = 25$ m, $x = 75$ m
½ mark
$h = 253$ m ( $\approx 43.3$ m). So each pole is $253$ m high; the point is 25 m from the pole whose top is seen at $6 0^{\circ}$ and 75 m from the other pole.
½ mark

Answer: Height of each pole = $253$ m (≈ 43.3 m); distances of the point from the poles: 25 m and 75 m

Where the marks are

Correct labelled figure (1 mark)
$tan 6 0^{\circ} = h / y$ giving $h = 3 y$ (1 mark)
$tan 3 0^{\circ} = h / x$ giving $h = x / 3$ (1 mark)
Obtains $x = 3 y$ with $x + y = 100$ (1 mark)
$x = 75$ m, $y = 25$ m (½ mark)
$h = 253$ m (½ 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 paper2024 board paper, set 430/3/1 (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 2024: question 34

Hints

Step-by-step solution

Revise the chapter

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