MathsCBSE Class 10 · Basic · Board exam 2023, set 430/4/1 · 5 marks · Long answer

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

Set in the CBSE Class 10 board exam of 2023, paper 430/4/1, 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

The shadow of a tower standing on a level ground is found to be $40$ m longer when the Sun's altitude is $3 0^{\circ}$ than when it was $6 0^{\circ}$ . Find the height of the tower.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/4/1.

Hints

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

Hint 1

Draw the tower and two shadows: the shadow at 30° is 40 m longer than the shadow at 60°.

Hint 2

Write tan 60° and tan 30° in the two right triangles and eliminate x.

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: tower AB = $h$ m with foot A; C and D on the ground on the same side, $A C = x$ m (shadow at $6 0^{\circ}$ ), $C D = 40$ m, $∠ A C B = 6 0^{\circ}$ , $∠ A D B = 3 0^{\circ}$ .
1 mark
In right $△ B A C$ : $tan 6 0^{\circ} = \frac{h}{x} \Rightarrow 3 = \frac{h}{x} \Rightarrow h = 3 x$ ...(1)
1½ marks
In right $△ B A D$ : $tan 3 0^{\circ} = \frac{h}{x + 40} \Rightarrow \frac{1}{3} = \frac{h}{x + 40} \Rightarrow x + 40 = 3 h$ ...(2)
1½ marks
Substituting (1) in (2): $x + 40 = 3 (3 x) = 3 x \Rightarrow 2 x = 40 \Rightarrow x = 20$ m
½ mark
$h = 203$ m. Height of the tower is $203$ m (≈ 34.64 m).
½ mark

Answer: Height of tower = 20√3 m (≈ 34.64 m)

Where the marks are

Correct figure (1 mark)
$tan 6 0^{\circ} = \frac{h}{x}$ giving $h = 3 x$ (1½ marks)
$tan 3 0^{\circ} = \frac{h}{x + 40}$ giving $x + 40 = 3 h$ (1½ marks)
Finds $x = 20$ m (½ mark)
Height $= 203$ m (½ mark)

Revise the chapter

Chapter: Some Applications of Trigonometry (notes, formulas, where marks are lost and every past question)
Topic: Problems with Two Right Triangles

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 430/4/1 (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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