MathsCBSE Class 10 · Standard · Board exam 2023, set 30/6/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2023: question 37

Set in the CBSE Class 10 board exam of 2023, paper 30/6/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedSome Applications of Trigonometry

Radio towers are used for transmitting a range of communication services including radio and television. The tower will either act as an antenna itself or support one or more antennas on its structure. On a similar concept, a radio station tower was built in two Sections A and B. Tower is supported by wires from a point O.

Distance between the base of the tower and point O is 36 cm. From point O, the angle of elevation of the top of the Section B is $3 0^{\circ}$ and the angle of elevation of the top of Section A is $4 5^{\circ}$ .

Based on the above information, answer the following questions:

(i)
Find the length of the wire from the point O to the top of Section B.
1 mark
(ii)
Find the distance AB.
2 marks
OR
(ii)
Find the area of $△ O P B$ .
2 marks
(iii)
Find the height of the Section A from the base of the tower.
1 mark

Figure in the paper: Vertical tower PA on the left with base P on the ground. Section B is the lower part with top at B on the tower, Section A the upper part with top at A. Point O is on the ground 36 cm from P. Wires run from O to A and from O to B. Angle of 45° is marked at O between OP and OA; angle of 30° is marked at O between OP and OB. Segment PO = 36 cm is labelled on the ground. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 30/6/1.

Hints

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

Hint 1

Triangles OPB and OPA are right-angled at P, and OP = 36 cm is common to both.

Hint 2

The wire OB is the hypotenuse, so use cos 30°. For the heights PB and PA, use tan of the angle at O.

Hint 3

AB is the part of the tower between B and A, so subtract PB from PA.

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.

(i) In right $△ O P B$ (right-angled at P), $cos 3 0^{\circ} = \frac{O P}{O B}$ .
½ mark
(i) $\frac{3}{2} = \frac{36}{O B}$ , so $O B = \frac{72}{3} = 243$ cm.
The wire from O to the top of Section B is $243$ cm long ( $\approx 41.57$ cm).
½ mark
(ii) In $△ O P B$ , $tan 3 0^{\circ} = \frac{P B}{O P}$ .
$\frac{1}{3} = \frac{P B}{36}$ , so $P B = \frac{36}{3} = 123$ cm.
1 mark
(ii) In right $△ O P A$ , $tan 4 5^{\circ} = \frac{A P}{O P}$ .
$1 = \frac{A P}{36}$ , so $A P = 36$ cm.
½ mark
(ii) $A B = A P - P B = 36 - 123 = 12 (3 - 3)$ cm ( $\approx 15.22$ cm).
½ mark
(ii) OR: Area of $△ O P B = \frac{1}{2} \times O P \times P B$ , where $P B = 36 tan 3 0^{\circ} = 123$ cm.
1 mark
(ii) OR: Area $= \frac{1}{2} \times 36 \times 123 = 2163$ cm $^{2}$ ( $\approx 374.1$ cm $^{2}$ ).
1 mark
(iii) The height of Section A from the base is $A P$ . In $△ O P A$ , $A P = 36 tan 4 5^{\circ} = 36$ cm.
1 mark

Answer: (i) OB = 24√3 cm; (ii) AB = 12(3 − √3) cm OR area = 216√3 cm²; (iii) 36 cm

Where the marks are

Uses $cos 3 0^{\circ} = O P / O B$ (½ mark)
Gets $O B = 243$ cm (½ mark)
Finds $P B = 123$ cm using $tan 3 0^{\circ}$ (1 mark)
Finds $A P = 36$ cm using $tan 4 5^{\circ}$ (½ mark)
Gets $A B = 36 - 123$ cm (½ mark)
OR part: writes the area formula with $P B = 123$ cm (1 mark)
OR part: area $= 2163$ cm $^{2}$ (1 mark)
Height of Section A, $A P = 36$ cm (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 30/6/1 (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 37

Hints

Step-by-step solution

Revise the chapter

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