1. a vertical post stands on a horizontal plane. the angle of elevation of the top is 60 ° and that of a point xm be the height of the post, then prove that x=2h/3.

2. a fire in a building B is reported on telephone to 2 fire stations P nd Q, 10 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 angle of 45 ° to the road. which station shold send its team and how much will the team have to travel?

3.a man on a top of tower observes a truck at angle of depressionα where tan α= 1/√5 and sees that it is moving towards the base of the tower. ten minutes later, the angle of depression of the truck is found to be β where tan β= √5, if the truck is moving at a uniform speed, determine how much more time it will take to reach the base of the tower.

Let AB be the vertical tower. Suppose D and C be the positions of the truck when the angle of depression from the top of the tower is ∝ and β respectively.

Suppose the uniform speed of the truck be v m/min.

Time taken for the angle of depression to change from ∝ to β = 10 min  (Given)

∠EAD = ∠ADB = ∝  (Alternative angles)

∠EAC = ∠ACB = β  (Alternative angles)

Suppose AB = h m and BC = x m

CD = Distance covered by car in 10 min = v m / min × 10 min = 10v m

In ΔADB,

In ΔACB,

From (1) and (2), we get

Time taken by truck to reach the tower from

∴ Time taken by truck to reach the tower C is 2.5 min.

 

2.

Let AB be the building. P and Q are two fire stations.

Given, PQ = 10 km, ∠BPA = 60° and ∠BQA = 45°.

In ΔABP,

In ΔABQ,

From (1) and (2)

The fire station P is nearer to the building, therefore, team from station P should be sent to the building.

AP + AQ = 10 km  (Given)

Thus, distance travelled by the team from station P is .

 

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