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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?
Solution
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
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