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Example –3. From the top of a hill, the angles of depression of two consecutive kilometer stones, due east are found to be 30ºand 45º respectively. Find the distance of the two stones from the foot of the hill.

Solution :-

Fig. – 5.

Let AB be the hill, B be the foot of the hill and C and D the two km stones.
                 Let AB = h and BC = x.  
                 Hence    DC = 1 km = 1000 m
                 In Δ ABC,  tan 45º = AB/BC
                 Or,                        1 = h/x 
                 Or,                         h = x      (i)……………..

                  In t ABD,    tan30º  = AB/BD
                                           1/√3 = h/(x + 1000)
                                    x + 1000 = h√3
                                    x + 1000 = x√3   [from (i), h = x]
                                     x(√3 – 1) = 1000
                                                x  = 1000/(√3 – 1)
                                                    =  1000 (√3 + 1)/(√3 – 1)(√3 + 1)
                                                    =  1000 (√3 + 1)/2
                                                    =  500 (1.732 + 1)
                                                    =  500 × 2.732 = 1366 m = 1.366 km.

Thus, first km stone is 1.366 km and second km stone is 2.366 km.