Krishna
0

Step 1: Read the given question and make a figure using the given information.

            

            GIVEN:  1.2 m girl spots a balloon

                          Girl height AG = BF = 1.2 m ( \because AC // GF)

                          Height of the balloon EF = 88.2 m

                         The angle of elevation of the balloon from the eye of the girl

                                         \angle EAB = 60\degree

                         After some time, angle of elevation

                                         \angle DAC = 30\degree

                        Horizontal distance traveled by the balloon BC = ?


                FROM THE FIGURE:

                      Height of balloon above the girl height 

                          BE = EF - BF

                                =  88.2 - 1.2

                          BE = 87 m

                          BE = DC = 87 m................(1)

                                                                                                            

Step 2: Find the possible distances between the points using the trigonometric ratios

                  From right triangle ABE

                      \tan\theta=\frac{opposite\ side\ of\ the\ angle}{adjacent\ side\ of\ the\ angle}


                       \tan 60\degree = \frac{87}{AB}


                       \sqrt{3} = \frac{87}{AB}              \because \tan 60\degree = \sqrt{3}


                       AB = \frac{87}{\sqrt{3}}

                      

                  From right triangle  ADC

                         \tan A = \frac{CD}{AC} = \frac{87}{AC}


                         \tan 30\degree = \frac{87}{AC}


                         \frac{1}{\sqrt{3}} = \frac{87}{AC}                             \because \tan 30\degree = \frac{1}{\sqrt{3}}


                         AC = 87\sqrt{3}

                        

Step 3: Find the horizontal distance traveled by the balloon

              From the figure:  

                            AC = AB + BC

                            BC = AC - AB                  

                            BC = 87\sqrt{3} - \frac{87}{\sqrt{3}}


                           BC = 87 (\sqrt{3} - \frac{1}{\sqrt{3}})


                           BC = 87 (\frac{\sqrt{3}*\sqrt{3} - 1}{\sqrt{3}})


                          BC=87\left(\frac{3-1}{\sqrt{3}}\right)


                           BC = \frac{87*2}{\sqrt{3}}


                      Multiply \sqrt{3} numerator and denominator  


                            BC = \frac{87 * 2*\sqrt{3}}{\sqrt{3}*\sqrt{3}}


                            BC = \frac{87*2*\sqrt{3}}{3}


                           BC = 29*2*\sqrt{3}


                           BC = 58\sqrt{3}

              Hence, distance traveled by the balloon BC = 58\sqrt{3}