Q) Two pillars of equal lengths stand on either side of a road which is 100 m wide, exactly opposite to each other. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are 60° and 30°. Find the length of each pillar and distance of the point on the road from the pillars. (Use √3 = 1·732)

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 100 m

∴ P + Q = 100

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

∴ 3 H + H = 100 √3

∴ 4 H = 100 √3

∴ H = 25 √3 m = 25 x 1.732

∴ H = 43.30 m 

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

∴ P = 25 √3 x √3

∴ P = 75 m

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

∴ Q =

∴ Q = 25 m

Therefore, height of the poles is 43.30 m and the distance of point from poles is 75 m and 25 m.

Check: We just calculated that, P  = 75 m and Q = 25 m, therefore width of the road = 75 + 25  = 100 m

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

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