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Trig Word Problem

Date: 5/5/96 at 23:13:27
From: marg atienza
Subject: Trig problem

An engineer determines that the angle of elevation from her position 
to the top of s tower is 52 degrees. She measures the angle of 
elevation again from a point 47m farther from the tower and finds it 
to be 31 degrees. Both positions are due east of the town. Find the 
height of the tower.
I hope you can help.  Thanks.

Date: 6/21/96 at 16:35:35
From: Doctor Patrick
Subject: Re: Trig problem

Hi!  This would be a fairly easy problem if we knew how far the person 
was standing from the tower - all we would have to do is use some 
basic trig.

If we made a right triangle with the height of the tower as one leg, 
and the ground as the other, then the height(h) over the distance 
along the ground would be equal to the tangent of the angle from the 
ground to the top of the tower, since the tan function is equal to the 
opposite side (the height) over the adjacent side (the distance along 
the ground).

  | \
h |  \
  |   \
  |    \
  ------* <-- angle from where the observer is (52 degrees).
(distance along ground)   

All you would have to do is to fill in the numbers you knew - the 
tangent of the angle and the distance you were from the tower - and 
you could solve for the height with the formula h/x=tan(52).

Unfortunately, it's not that easy. We don't know how far we are, so we 
move back another 47m to get more information to solve the problem 
with.  For now, let's call that unknown distance along the ground x, 
and the height of the tower h.

We find that at that distance (x+47m) the angle to the top of the 
tower is only 31 degrees.  Now we know that the tangent of 31 degrees 
is equal to the height of the tower over x (the original distance) 
plus the additional 47m.  Our formula is h/(x+47) = tan(31).  Do you 
see how I got that?  It's the same trig function we used above, taken 
from the new distance.

Now we have two equations and two variables (h and x).  Let's see if 
we can find an answer using both of them.  Our two equations are:

  (1) from the first point -  h/x = tan(52)
  (2) from the second point - h/(x+47) = tan(31)

Do you see any ways we could put these together?  

The first thing that came to my mind was to rewrite both equations to 
get h on one side of the equals sign and the rest of the numbers on 
the other. This makes our equations

  (1) h = tan(52)*x
  (2) h = tan(31)*(x+47)

If we substitute for the "h"s then we get tan(52)*x=tan(31)*(x+47).  
Using my calculator I find that tan(53)=1.2799416 and tan(31)=

Plugging those numbers into the equation we get 
1.27994168*x = 0.60086062*(x+47)

We can then rewrite that as 1.2799416x = 0.60086062x+28.240449.

Moving the "x"s to one side we get 0.65507228x=28.240449.  Then we 
solve for x.  X = 28.240449/0.65507228 = 41.586274m. 

Now that we have the x term, we can plug it into either equation 
(though I'd choose the first since it's simpler) and solve for h.  
Do you think you can handle that part on your own?  If not, write back 
and I'll give you a hand.

-Doctor Patrick,  The Math Forum
 Check out our web site!   
Associated Topics:
High School Trigonometry

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