SOLUTION: From the top of a hill 42.5 m above a stream, the angles of depression to a point on the near shore and a point on the opposite shore are 42.3° and 40.6°, respectively. Find the

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Question 1149147: From the top of a hill 42.5 m above a stream, the angles of depression to a point on the near shore and a point on the opposite shore are 42.3° and 40.6°, respectively. Find the width of the stream between these two points. Include a diagram.
Found 2 solutions by ikleyn, jim_thompson5910:
Answer by ikleyn(52878) About Me  (Show Source):
You can put this solution on YOUR website!
.

The longer distance is  42.5%2Ftan%2840.6%5Eo%29.


The shorter distance is  42.5%2Ftan%2842.3%5Eo%29.


The width of the stream is the difference of these distances.


Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

The diagram looks like this

List of points
A = location of the surveyor at the top of the hill
B = location directly below point A
C = location on the near shore
D = location on the opposite shore

We know that AB = 42.5 as this is given to us. What we want to find is the distance from C to D, or the length of segment CD. Let's call that x for now. We'll also need to know the distance from B to C, so let's call that y.

x = length of CD = width of stream
y = length of BC
x+y = length of BD

We can split up the triangles to get this

and this


Triangle ABC lets us see that
tan(angle) = opposite/adjacent
tan(angle ACB) = AB/BC
tan(42.3) = AB/BC
tan(42.3) = 42.5/y
y*tan(42.3) = 42.5
y = 42.5/tan(42.3)
y = 46.7068901477929
note: make sure your calculator is in degree mode, the result above is approximate

Triangle ABD lets us say,
tan(angle) = opposite/adjacent
tan(angle ADB) = AB/BD
tan(40.6) = AB/BD
tan(40.6) = 42.5/(x+y)
(x+y)*tan(40.6) = 42.5
x*tan(40.6)+y*tan(40.6) = 42.5
x*tan(40.6) = 42.5-y*tan(40.6)
x = (42.5-y*tan(40.6))/tan(40.6)
x = (42.5-46.7068901477929*tan(40.6))/tan(40.6)
x = 2.87871067288322
x = 2.9
I'm rounding to one decimal place as the given distance 42.5 meters is also rounded to one decimal place.

The approximate answer is 2.9 meters

Edit: The solution provided by the tutor @ikleyn is much more simple, so it might be better to go with that method.