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Question: Two cars are on the same side of a spire of height h ft. The angles of depression of these two cars from the top of the spire are φ and θ. Calculate the distance between the cars when h = 100, φ = 25° and θ = 40°.


A ) 96 ft
B ) 81 ft.
C ) 334 ft.
D ) 20 ft

Steps to derive

4 The height of the spire is AB = h = 100 ft, P and Q are the positions of the cars , and let PQ = x ft, AP = y ft as shown.
 [Draw the diagram for the given data.]


5 The angle of depressioin of Q at B = φ and the angle of depression of P at B = θ

6  BPA = θ, BQA = φ
 [Alternate angles are equal.]

7 In the right triangle BQA, tan φ = tan 25° = 100y and hence y » 215
 [Find y from the right triangle BQA.]

8 In the right triangle BAP, tan θ = tan 40° = 100APand hence AP » 119.189
 [Find AP from the right triangle BAP.]

9 AP » 119.189
 [Substitute the value of tan 40° and simplify.]

10 AP + x = y
 [Use AP + PQ = AQ.]

11 x = y - AP = 215 - 119.189 » 96
 [Find x.]

12 Therefore, the distance between the cars is 96 ft.



Hence the right answer is Option A

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