Question

A fire in a building B is reported on the telephone to two fire stations P and Q, 20 km apart from each other on a straight road. P observes that the fire is at an angle of 60° to the road and Q observes that it is at an angle of 45° to the road. Which station should send its team and how much will this team have to travel?

Answer 1

Let AB be the building of height h. P Observes that the fire is at an angle of 60° to the road and Q observes that the fire is at an angle of 45° to the road.

Let QA = x, AP = y. And  `∠BPA = 60^@`,∠BQA = 45°, given PQ = 20.

Here clearly ∠APB > ∠AQB

=> ∠ABP < ∠ABQ

=> AP < AQ

So station P is near to the building. Hence station P must send its team

We sketch the following figure

So we use trigonometric ratios.

In ΔPAB

`tan P = (AB)/(AP)`

`=> tan 60^@ = h/y`

`=> h = sqrt3y`

Again in ΔQAB

`=> tan Q = (AB)/(QA)`

`=> tan  45^@  = h/x`

`=> 1 = h/x`

`=> x = h`

Now

x + y = 20

`=> h + y = 20`        [∵ x = h]

`=> sqrt3y + y = 20`      [∵ `h = sqrt3y`]

`=> y = 20/(sqrt3 + 1) = 10(sqrt3 - 1)`

Hence the team from station P wil have to travel `10(sqrt3 - 1)` km