MathsCBSE Class 10 · Basic · Board exam 2023, set 430/2/1 · 5 marks · Long answer

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

Set in the CBSE Class 10 board exam of 2023, paper 430/2/1, section D, question 32(b), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(b) · 5 marks · Long answerCompetency-basedSome Applications of Trigonometry

The angle of elevation of the top of a building from a point A on the ground is $3 0^{\circ}$ . On moving a distance of 30 m towards its base to the point B, the angle of elevation changes to $4 5^{\circ}$ . Find the height of the building and the distance of its base from point A. (Use $3 = 1 \cdot 732$ )

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/2/1.

Hints

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

Hint 1

Let BC = x; then AC = x + 30. Draw two right triangles sharing the building as the vertical side.

Hint 2

From the 45° angle you get a simple relation between h and x.

Hint 3

Substitute it into the tan 30° equation and rationalise the denominator.

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 CD = h m be the building with foot C and top D. A and B are points on the ground in line with C, AB = 30 m, BC = x m, so AC = x + 30 m. ∠DAC = 30°, ∠DBC = 45°, ∠DCA = 90°.
1 mark
In right $△ D C A$ : $tan 3 0^{\circ} = \frac{h}{x + 30} \Rightarrow \frac{1}{3} = \frac{h}{x + 30}$ ...(i)
1½ marks
In right $△ D C B$ : $tan 4 5^{\circ} = \frac{h}{x} \Rightarrow h = x$ ...(ii)
1 mark
Putting (ii) in (i): $3 h = h + 30 \Rightarrow h (3 - 1) = 30 \Rightarrow h = \frac{30}{3 - 1} \times \frac{3 + 1}{3 + 1} = 15 (3 + 1)$
½ mark
$h = 15 \times 2.732 = 40.98$ m. Height of the building = 40.98 m.
½ mark
Distance of the base from A $= x + 30 = 40.98 + 30 = 70.98$ m.
½ mark

Answer: Height of building = $15 (3 + 1)$ = 40.98 m; distance of base from A = 70.98 m

Where the marks are

Correct labelled figure (1 mark)
Forms $tan 3 0^{\circ} = \frac{h}{x + 30}$ (1½ marks)
Forms $tan 4 5^{\circ} = \frac{h}{x}$ , so h = x (1 mark)
Solves to $h = 15 (3 + 1)$ (1/2 mark)
Height = 40.98 m (1/2 mark)
Distance from A = 70.98 m (1/2 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 430/2/1 (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(b)

Hints

Step-by-step solution

Revise the chapter

More questions from Some Applications of Trigonometry