River, triangle

Question

The angles of depression To the near and far banks of a river measure 49 degrees and 11 degrees respectively. If the observer's eyes are 5ft above the ground, how wide is the river? 

Answer 1

It helps to draw things, but that's impossible to do here, so I'll describe it as best I can.

 

The first thing we have to do is determine where the observer is standing.  Let's let x be the number of feet away from the near bank that he is standing.  His eyes, the ground under his feet, and the near bank form a right triangle, where the right angle is at his feet.  One leg of the triangle the height of his eyes, 5.  We also know the angle at his eyes (that is the angle from the near bank to his eyes to his feet).  Since the angle of depression to the near bank is 49, the angle at this point of the triangle is 90 - 49 = 41.

 

So that gives us the equation tan 41° = x / 5.

 

So x = 5 tan 41° = 4.36 feet.  That's how far from the river he's standing.

 

Now we form another right triangle, from his eyes to his feet to the far bank.  One leg is 5 again (from the ground to his eyes).  The other leg is the width of the river (call that w) plus his distance from the near bank (4.36) -- w + 4.36.

 

And again, since the angle of depression to the far bank is 11°, the angle at his eyes is 90 - 11 = 79°.

 

So now we have another equation that we can solve: 

 

(w + 4.36) / 5 = tan 79°

w + 4.36 = 5 tan 79°

w + 4.36 = 25.72

w = 21.36

 

So the width of the river is 21.36 feet.