Question

A vertical pole and a vertical tower are on the same level ground in such a way that from the top of the pole, the angle of elevation of the top of the tower is 60° and the angle of depression of the bottom of the tower is 30°. Find:

1. the height of the tower, if the height of the pole is 20 m;
2. the height of the pole, if the height of the tower is 75 m.

Answer 1

Let AB be the tower and CD be the pole. 

Then ∠ACE = 60° and ∠BCE = 30°.

i. In ΔBEC, 

`(BE)/(EC) = tan 30^circ`   

`=> 20/(EC) = 1/sqrt(3)` 

`=> EC = 20sqrt(3)  m` 

In ΔAEC, 

`(AE)/(EC) = tan 60^circ` 

`=> AE = 20sqrt(3) xx sqrt(3) = 60  m `  

∴ Height of the tower = AB

= AE + EB 

= (60 + 20)

= 80 m

ii. Let height of the pole be x m

∴ CD = BE = x 

In ΔBEC, 

`(BE)/(EC) = tan 30^circ` 

`=> EC = sqrt(3)x` 

In ΔAEC,

`(AE)/(EC) = tan 60^circ` 

`=> (75 - x)/(EC) = sqrt(3)` 

`=>` 75 – x = 3x

∴ `x = 75/4 = 18.75  m`

∴ Height of the pole is 18.75 m.