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.

 

Case Study Questions

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 stations A and B (B vertically below A). The tower is supported by wires AO and BO from a point O on the ground. Distance between the base C of the tower and the point O is 36 m. From O, the angles of  elevation of the tops of station B and station A are 30º and 45º respectively.

Based on the above, answer the following questions :

(i) Find the height of station B.

(ii) Find the height of station A.

(iii) Find the length of the wire OA.

(iv) Find the length of the wire OB. (CBSE 2022 Set 2)

Solution:

Look at the figure, which we construct now

(i)  Find the Height of  station B

So, we need to find the length of BC

We can take tan O

Tan 30⁰ =

We know the value of tan 30 is 1/Ö3.  Substitute the values

        Cross multiply, we get

        BC =  =  = 12Ö3 m

Hence the length of tower B is 12Ö3 m.

(ii) Find the height of the station A

Here we want to find the length of AC.

CO is known and the angle is 45⁰.  So we take tan 45

Tan 45⁰    =

tan 45⁰     =

We know that tan45⁰ = 1, so substitute the values

        1       =

        AC = 36 m

Hence the height of tower A is 36 m.

(iii)  Find the length of OA

OA is the hypotenuse of the triangle AOC.

Here we know that the length of CO is 36 m and length of AC is also 36 m.

Here we take sin 45⁰.

i.e.,

We know that sin45⁰ = 1/Ö2.  Substitute the values

       

Cross multiply, we get

        AO = 36Ö2 m

Hence the length of the wire AO is 36Ö2 m.

(iv)  Find the length of the wire OB

Consider the triangle BCO.  Here BC and

OC are known and we want to find OB.  In this case we take cos 30⁰ .

        Cos 30⁰ =

We know that cost 30⁰  = .  Substitute the values, we get

       

Cross multiply we get,

        BO =  =  = 12 x 2 x Ö3

        BO = 24 Ö3 m

Hence the length of the wire BO is 24Ö3 m

Summary of Answers:

(i) Height of station B = 12Ö3 m

(ii) Height of station A  = 36 m

(iii) Length of the wire OA = 36Ö2 m

(iv) Length of the wire OB = 24Ö3 m