Q) Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 60° and 30°, respectively. Find the height of the poles and the distances of the point from the poles.                          

Ans:

Let’s start from the diagram for the question:

Let ‘s take AB and DE be the 2 poles. P is the distance of C from B and Q is the distance from E

Step 1: Let’s start from In Δ ABC, tan 30 =

∴ tan 30⁰ =

∴ P = H √3 …………. (i)

Step 2: Next, in Δ CDE, tan 60 =

∴ tan 60⁰ =

∴

∴ Q =   …………. (ii)

(Note: Here we calculate P and Q in terms of H. When we will get all H terms together and value of H will be calculated.)

Step 3: Given that the width of road is 80 m

∴ P + Q = 80

By substituting, value of P and Q from equation (i) in equation (ii), we get:

∴ 3 H + H = 80 √3

∴ 4 H = 80 √3

∴ H = 20 √3 m

Step 4: From equation (i), we have P = H √3

∴ P = 20 √3 x √3

∴ P = 60 m

Step 5: From equation (ii), we have Q =

∴ Q =

∴ Q = 20 m

Therefore, height of the poles is 20 √3 m and the distance of point from poles is 60 m and 20 m.

Check: We just calculated that, P  = 60 m and Q = 20 m, therefore width of the road = 60 + 20  = 80 m

Since this matches with given data in the question, hence our calculation is correct.

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