MathsCBSE Class 10 · Basic · Board exam 2023, set 430/5/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 32

Set in the CBSE Class 10 board exam of 2023, paper 430/5/1, section D, question 32, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32 · 5 marks · Long answerCompetency-basedSome Applications of Trigonometry

A person walking 48 m towards a tower in a horizontal line through its base observes that angle of elevation of the top of the tower changes from $4 5^{\circ}$ to $6 0^{\circ}$ . Find the height of the tower and distance of the person, now, from the tower. (Use $3 = 1.732$ )

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/5/1.

Hints

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

Hint 1

Draw the tower and mark the two observation points 48 m apart; the nearer point has the larger angle.

Hint 2

Let the present distance be x; write tan 45° and tan 60° in terms of h and x.

Hint 3

Equate the two expressions for h and rationalise $\frac{48}{3 - 1}$ .

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 = $h$ m be the tower (A at the foot, B the top). D is the first position of the person and C the new position, with D, C, A on a horizontal line, DC = 48 m and CA = $x$ m. $∠ B D A = 4 5^{\circ}$ , $∠ B C A = 6 0^{\circ}$ , $∠ B A D = 9 0^{\circ}$ .
1 mark
In right $△ B A D$ : $tan 4 5^{\circ} = \frac{A B}{A D} = \frac{h}{x + 48} \Rightarrow h = x + 48$ …(i)
1 mark
In right $△ B A C$ : $tan 6 0^{\circ} = \frac{A B}{A C} = \frac{h}{x} \Rightarrow h = 3 x$ …(ii)
1½ marks
From (i) and (ii): $3 x = x + 48 \Rightarrow x = \frac{48}{3 - 1} = \frac{48 ( 3 + 1 )}{2} = 24 (3 + 1) = 24 \times 2.732 = 65.57$ m
1 mark
$h = x + 48 = 65.57 + 48 = 113.57$ m
½ mark

Answer: Height of tower ≈ 113.57 m; present distance from tower ≈ 65.57 m

Where the marks are

Correct labelled figure (1 mark)
$tan 4 5^{\circ}$ gives $h = x + 48$ (1 mark)
$tan 6 0^{\circ}$ gives $h = 3 x$ (1½ marks)
Solves $x = 24 (3 + 1) = 65.57$ m (1 mark)
$h = 113.57$ 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 paper2023 board paper, set 430/5/1 (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 2023: question 32

Hints

Step-by-step solution

Revise the chapter

More questions from Some Applications of Trigonometry