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To determine the width of a​ river, forestry workers place markers on opposite sides of the river at points A and B. A third marker is placed at point​ C, 45 meters away from point​ A, forming triangle ABC. If the angle in triangle ABC at point C is 55 degrees and the angle in triangle ABC at point A is 117 degrees​, then determine the width of the river to the nearest tenth of a meter.

Respuesta :

Answer:

x = 6,8 m

Step-by-step explanation: See attached figure

We have to find   the river width   CD =  x

In triangle ABC  and by law of sin

Sum of inside angles in a triangle  =  180⁰

Then    ∠ ABC  + ∠CAB  + ∠BCA    =  180⁰

∠ ABC  =  55⁰

∠CAB   =  117⁰

Then     ∠BCA  = 8⁰

We apply angles sinus law  in that triangle to get b

45/sin55⁰  = b/ sin8⁰                   sin 55⁰  =  0,819152    from tables

                                                     sin 8 ⁰   =  0,139173

                                                           b  =  45  m

By subtitution

45/0,819152    = b / 0,139173

b  =   45* 0,139173  /0,819152

b  =  7,6454  m

Now we change to triangle BAD   and apply again angles sinus law

b/sin90⁰  =  x/sin 63                                   sin 90⁰   =  1      b  = 7,6454  m

                                                                    sin 63⁰   =  0,891007

By subtitution

7,6454/ 1  =  x / 0,891007

x = 7,6454*0,891007

x = 6,8 m

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