SOLUTION: Standing on the bank of a river, a surveyor measures the angle to the top of a tree on the opposite bank to be 23 degrees. He backs up 45ft and remeasures the angle to the top of

Question 180794This question is from textbook Analytic Trigonometry
: Standing on the bank of a river, a surveyor measures the angle to the top of a tree on the opposite bank to be 23 degrees. He backs up 45ft and remeasures the angle to the top of the tree at 18 degrees. How wide is the river? This question is from textbook Analytic Trigonometry

Found 2 solutions by stanbon, josmiceli:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Standing on the bank of a river, a surveyor measures the angle to the top of a tree on the opposite bank to be 23 degrees. He backs up 45ft and remeasures the angle to the top of the tree at 18 degrees. How wide is the river?
-------------------------------------
Let the width of the river be "x".
Draw the picture and solve for "x".
---------------------------
tan(23) = tree/x
tan(18) = tree/(x + 45)
-----------------------
tree = x*tan(23)
tree = (x+45)*tan(18)
------------------------
Equation:
x*tan(23) - (x+45)*tan(18)
xtan(23) - x*tan(18) = 45*tan(18)
x[tan(23)-tan(18)] = 45*tan(18)
Solve for "x"
===========================
Cheers,
Stan H.
Answer by josmiceli(19441) (Show Source): You can put this solution on YOUR website!
Let the width of the river = ft
Let the height of the tree = ft
given:
(1)
(2)
These are 2 equations with 2 unknowns, so
they should be solvable

(1)
(2)
----------------------------
Multiply both sides of (1) by
Multiply both sides of (2) by
(1)
(2)
(2)
Subtract (2) from (1)
(1)
(2)

The river is 146 ft 9 in wide
check answer:
(2)

ft the height of the tree
(1)

close enough
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