MathsCBSE Class 10 · Standard · Board exam 2023, set 30/1/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2023: question 33(a)

Set in the CBSE Class 10 board exam of 2023, paper 30/1/1, section D, question 33(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(a) · 5 marks · Long answerCompetency-basedSome Applications of Trigonometry

The angle of elevation of the top of a tower 24 m high from the foot of another tower in the same plane is $6 0^{\circ}$ . The angle of elevation of the top of second tower from the foot of the first tower is $3 0^{\circ}$ . Find the distance between two towers and the height of the other tower. Also, find the length of the wire attached to the tops of both the towers.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/1/1.

Hints

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

Hint 1

Draw both towers on the same ground line. The 60° angle is at the foot of the second tower, looking up at the 24 m tower.

Hint 2

Use $tan 6 0^{\circ}$ with the 24 m tower to find the gap, then $tan 3 0^{\circ}$ to find the second tower's height.

Hint 3

For the wire, use Pythagoras: the vertical difference is $24 - h$ and the horizontal distance is the gap.

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 $A B$ be the first tower with $A B = 24$ m (foot A, top B). Let $C D = h$ be the second tower (foot C, top D). Let $A C = x$ be the distance between the towers.
- $∠ B C A = 6 0^{\circ}$ (elevation of B from C)
- $∠ D A C = 3 0^{\circ}$ (elevation of D from A)
Draw $D E ⊥ A B$ (E on AB), so $D E = x$ and $A E = h$ .
1 mark
In right $△ D C A$ : $tan 3 0^{\circ} = \frac{C D}{A C} \Rightarrow \frac{1}{3} = \frac{h}{x} \Rightarrow x = h 3$ ...(i)
1 mark
In right $△ B A C$ : $tan 6 0^{\circ} = \frac{A B}{A C} \Rightarrow 3 = \frac{24}{x} \Rightarrow x = \frac{24}{3} = 83$ ...(ii)
1 mark
From (i) and (ii): the distance between the towers is $x = 83$ m (≈ 13.86 m).
The height of the other tower is $h = \frac{x}{3} = 8$ m.
1 mark
The wire joins tops B and D. In right $△ B E D$ :
- $B E = 24 - 8 = 16$ m
- $E D = x = 83$ m
$B D = B E^{2} + E D^{2} = 256 + 192 = 448 = 87$ m (≈ 21.17 m).
1 mark

Answer: Distance $= 83$ m; height of other tower = 8 m; wire $= 87$ m

Where the marks are

Correct labelled figure (1 mark)
$tan 3 0^{\circ} = h / x$ giving $x = h 3$ (1 mark)
$tan 6 0^{\circ} = 24/ x$ giving $x = 83$ (1 mark)
$x = 83$ m and $h = 8$ m (½ + ½ mark)
Wire length $= 1 6^{2} + (83)^{2} = 87$ 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 paper2023 board paper, set 30/1/1 (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 2023: question 33(a)

Hints

Step-by-step solution

Revise the chapter

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