Question

The angle of elevation of the top of a chimney from the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. If the height of the tower is 40 metres, find the height of the chimney. 

According to pollution control norms, the minimum height of a smoke-emitting chimney should be 100 metres. State if the height of the above-mentioned chimney meets the pollution norms. What value is discussed in this question?

The angle of elevation of the top of a chimney from the foot of a tower is 60° and the angle of depression of the foot of the chimney from the top of the tower is 30°. If the height of the tower is 40 m, find the height of the chimney. According to pollution control norms, the minimum height of a smoke emitting chimney should be 100 m. State if the height of the above mentioned chimney meets the pollution norms. What value is discussed in this question?

Answer 1

Let PQ be the chimney and AB be the tower.

We have,

AB = 40 m, ∠APB = 30° and ∠PAQ = 60°

In ΔABP,

`tan 30^circ = (AB)/(AP)`

`⇒ 1/sqrt(3) = 40/(AP)`

`⇒ AP = 40sqrt(3)  m`

Now, in ΔAPQ, 

` tan 60^circ = (PQ)/(AP)`

`⇒ sqrt(3) = (PQ)/(40sqrt(3))`

`PQ = 40sqrt(3) xx sqrt(3)`

PQ = 120 m

So, the height of the chimney is 120 m.

Now, DQ = PQ – PD

= 120 m – 40 m

= 80 m

`AP = BD = 40sqrt(3)  m`

In ΔBDQ

`BQ = sqrt((DQ)^2 + (BD)^2)`

= `sqrt((80)^2 + (40sqrt(3))^2)`

= `sqrt(6400 + 4800)`

= `sqrt(11200)`

= `40sqrt(7)  m`

Thus, length of wire tied from the top of the chimney to the top of tower is `40sqrt(7)  m`.