Krishna
0

Step 1:  According to the given measurements construct an imaginary figure.

                

                The two poles of equal heights are AB and PQ say

                                                AB = PQ = H

                  A point between the poles  = D say

                  The distance between the two poles (AB & PQ) = BQ = 120 feet

                  If you say length of BD = h m

                  Then, the length of DQ = (120 - h) m

                  The angle of elevation


Step 2: Determine the height of the pole by using the trigonometric ratios.

            FROM THE FIGURE:  

                        Take right triangle ABD

                               \tan \theta = \frac{opposite}{adjacent} = \frac{AB}{BD}

                               \tan 60\degree = \frac{AB}{h}


                                     \sqrt{3} = \frac{H}{h}               \because \tan 60\degree = \sqrt{3}


                                       H = h \sqrt{3}.....................(1)


                      Take right triangle  PQD

                                 \tan 30\degree = \frac{PQ}{DQ}


                                         \frac{1}{\sqrt{3}} = \frac{H}{120 - h}                \because \tan 30\degree = \frac{1}{\sqrt{3}}

  

                                             H = \frac{120 - h}{\sqrt{3}} ....................(2)


Step 3:  Solve equation (1) and (2)

                     H = h \sqrt{3} .......................(1)

                     H = \frac{120 - h}{\sqrt{3}}.......................(2)

              From equation (1)&(2) we can write    

                           h\sqrt{3} = \frac{120 - h}{\sqrt{3}}


                   h\sqrt{3} * \sqrt{3} = 120 - h


                         3h + h = 120


                                   h = \frac{120}{4}


                                    h = 30


Step 4: Find the unknown lengths by substituting the 'h' value.

            From equation (1)

                   H = h \sqrt{3}

                   H = 30\sqrt{3}


        The length of the  DQ = 120 - h

                                      DQ = 120 - 30

                                      DQ = 90


            Therefore, Height of the pole H = 30\sqrt{3}

              The distances of the point from the poles are BD = 30 feet

                                                                                      DQ = 90 feet