From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are 30° and 45° respectively. If the bridge is at a height of 3 m from the banks, find the width of the river. (Use √(3) = 1.73)

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/6/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 32(a)

Set in the CBSE Class 10 board exam of 2023, paper 430/6/1, section D, question 32(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(a) · 5 marks · Long answerSome Applications of Trigonometry

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are $3 0^{\circ}$ and $4 5^{\circ}$ respectively. If the bridge is at a height of 3 m from the banks, find the width of the river. (Use $3 = 1.73$ )

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/6/1.

Hints

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

Hint 1

Draw the vertical from the bridge point to the river; it splits the width into two parts.

Hint 2

Use tan of each angle = 3 / (horizontal part) in the two right triangles, then add the parts.

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: P is the point on the bridge, PQ = 3 m is perpendicular to the river AB, with A and B the banks on opposite sides. Angles of depression 30° and 45° give $∠ P A Q = 3 0^{\circ}$ and $∠ P B Q = 4 5^{\circ}$ [alternate angles].
1 mark
In right $△ A P Q$ : $tan 3 0^{\circ} = \frac{P Q}{A Q} = \frac{3}{A Q}$
1 mark
$\frac{1}{3} = \frac{3}{A Q} \Rightarrow A Q = 33$ m
½ mark
In right $△ P B Q$ : $tan 4 5^{\circ} = \frac{P Q}{B Q} = \frac{3}{B Q}$
1 mark
$1 = \frac{3}{B Q} \Rightarrow B Q = 3$ m
½ mark
Width $A B = A Q + B Q = 33 + 3$
½ mark
$= 3 (1.73 + 1) = 3 \times 2.73 = 8.19$ m. Width of the river is 8.19 m.
½ mark

Answer: Width of river = $3 (3 + 1)$ m = 8.19 m

Where the marks are

Correct figure (1 mark)
tan 30° = 3/AQ (1 mark)
AQ = 3√3 m (½ mark)
tan 45° = 3/BQ (1 mark)
BQ = 3 m (½ mark)
AB = 3√3 + 3 (½ mark)
Width = 8.19 m (½ 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 paper2023 board paper, set 430/6/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2023: question 32(a)

Hints

Step-by-step solution

Revise the chapter

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