How High is the Building?

Date: 03/08/2002 at 18:35:08
From: Nancy
Subject: Angle of elevation

Melissa and her friends decide to fly their paper airplanes out of the 
window of a tall building (one that has windows that open). Delbert is 
part of the ground crew, and Janet is the launch director. Delbert 
sights the window at a 75-degree angle from the ground, then backs up 
25 ft and sights it again, this time at a 64-degree angle. How high is 
the window?

What is the formula for this problem and how do I work it out?

Thanks for the help,
Nancy

Date: 03/08/2002 at 19:15:06
From: Doctor Jeremiah
Subject: Re: Angle of elevation

Hi Nancy,

The first time he looks there is a triangle that looks like:

  +
  |\
  | \
  |  \
  h   \
  |    \
  |   75\
  +--x---+

And the second time there is a triangle that looks like:

  +
  |\
  | \
  |  \
  h   \
  |    \
  |   64\
  +-x+25-+

If you solve the first triangle for x and use that when you solve the 
second triangle for h, it will be easy to answer.

Try it, but write back if you get stuck.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/10/2002 at 21:43:09
From: Nancy
Subject: Angle of elevation

I still don't know what to do. I don't understand how to do the first 
part. I don't have the X and I don't know the formula or method to 
use.

Thanks,
Nancy

Date: 03/11/2002 at 01:25:33
From: Doctor Jeremiah
Subject: Re: Angle of elevation(I don't know)

Hi Nancy,

Any time both angles and distances are used, trignometry is involved.
There is a lot of stuff about trignometry in the Frequently Asked
Questions area and in the archives that would probably help you out. 
Check out these pages:

   Trigonometry Formulas
   http://mathforum.org/dr.math/faq/formulas/faq.trig.html   

   Basics of Trigonometry
   http://mathforum.org/dr.math/problems/coley.12.12.01.html   

So from the first triangle:

   +
   |\
   | \
   |  \
   h   \
   |    \
   |   75\
   +--x---+

We can get that the tan(75) = h/x, which means x tan(75) = h

And from the second triangle:

   +
   |\
   | \
   |  \
   h   \
   |    \
   |   64\
   +-x+25-+

We can get that the tan(64) = h/(x+25), which means (x+25) tan(64) = h

So we have these two equations:
       x tan(75) = h
  (x+25) tan(64) = h

Now, since both those equations are equal to h they must be equal to 
each other:

              x tan(75) = (x+25) tan(64)
              x tan(75) = x tan(64) + 25 tan(64)
  x tan(75) - x tan(64) = 25 tan(64)
  x (tan(75) - tan(64)) = 25 tan(64)
                      x = 25 tan(64) / (tan(75) - tan(64))

Which can be solved with a calculator. Then you take x tan(75) = h and 
stick in that value of x and solve for h.

Write back if you are still confused.

- Doctor Jeremiah, The Math Forum
  http://mathforum.org/dr.math/