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Determine Flagpole Height without Access to the Pole

Date: 12/01/2003 at 17:28:16
From: Ken
Subject: Calculate flagpole height without approaching within 10'

I teach English and computers at a community college, though I enjoy 
math problems.  A student came to me with a problem that, despite two 
days puzzling, still has me stumped.  The student has to calculate the 
height of the college's flagpole.  Normally, this problem would be 
simply solved by measuring the flagpole's shadow to get one leg of the 
right triangle formed by the flagpole and the earth.  The height could 
then be calculated using the trigonometric functions.

Here's the kicker, though:  The student can't approach the flagpole 
any closer than 10', and the student can't use any measuring device 
to determine the 10' limit (thus making impossible an accurate 
measurement of the flagpole's shadow).

I've tried approaching the problem using the trignometric functions, 
by analyzing the problem as a parallel-lines situation, and by using 
a mirror to create two similar triangles.  In all cases, I could 
learn much useful information, but in no case did I ferret out enough 
information to calculate the flagpole's height.

Date: 12/01/2003 at 21:16:41
From: Doctor Douglas
Subject: Re: Calculate flagpole height without approaching within 10'

Hi Ken.

Thanks for writing to the Math Forum.

This is a trigonometry problem, and while most students encounter
trigonometry in the context of right triangles, we can also apply it 
to general triangles as well:

  F.                     FOA is a right triangle.
  | . .                  The point O is inaccessible.
  |   .  .               We want to compute the height OF.
  |     .   .            We measure the distance BA and the
  |       .    .         angles of elevation OBF and OAF.
  |         .     .

The triangle of interest is FAB.  We have measured two angles of this 
triangle:  angle A (measured elevation from A), angle B (equal to 180 
deg - elevation from B), and angle F (180 deg - A - B).

Using the Law of Sines, we can write

  dist(BA)  dist(FA)
  ------- = -------        this allows us to calculate the
   sin(F)    sin(B)        distance FA.

And then we apply trigonometry to the right triangle FOA:

  dist(FO) = dist(FA)*sin(A),

where our final answer is expressed in terms of quantities that we 
have either measured directly or have computed from our measurements.  

One could carry out the angle measurements either by using a sighting 
instrument (such as a straw taped to a protractor with a weight 
suspended from the origin), or by marking the location of the flagpole 
shadow at the precise time that one measures the length of a shadow of 
some shorter object such as a yardstick.  The shorter object gives us 
information about the slope of the sun's rays via the arctangent. 

I hope this helps.  Please write back if you need more explanation 
about what is going on.

- Doctor Douglas, The Math Forum 

Date: 12/02/2003 at 06:30:48
From: Ken
Subject: Thank you (Calculate flagpole height without approaching
within 10')

I'm an English teacher with some interest (but no expertise) in Math.  
You, Doctor Douglas, obviously have such expertise; moreover, you
also write well (something to gladden an English teacher's heart on a
dreary December morning).  Thanks kindly for the help.

Associated Topics:
High School Trigonometry

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