# Pyramid

• Jan 16th 2013, 01:31 AM
Imonars
Pyramid
Hello guys, me and my friends are desperate.. We can't solve this, any ideas? Thanks!
http://i45.tinypic.com/140hx1t.png
• Jan 16th 2013, 02:44 AM
emakarov
Re: Pyramid
Introduce two notations and write two equations for the tangents of the angles shown.
• Jan 16th 2013, 02:49 AM
Imonars
Re: Pyramid
We have tried to find x , h and then solve for h
tan (40.3) = h / x + 200
tan (40.27) = h / x + 100
then we found x and solved for h but we got numbers like 0.525 which seem impossible to represent a high
• Jan 16th 2013, 02:59 AM
emakarov
Re: Pyramid
First, it's 46.27, not 40.27. Second, your equations are wrong because of the order of operations. Last, make sure that the tangent function interprets its arguments as given in degrees, not radians. The correct system gives the height between 440 and 470 feet.
• Jan 16th 2013, 03:07 AM
Imonars
Re: Pyramid
Sorry about the mistake, we have rearranged the equation to be: (x + 100)tan(46.27) = (x + 200)tan(40.3) we are trying to find the height
• Jan 16th 2013, 03:14 AM
emakarov
Re: Pyramid
Yes, this gives x(tan(46.27) - tan(40.3)) = 200 * tan(40.3) - 100 * tan(46.27), i.e.,

$\displaystyle x=\frac{200\tan40.3 - 100\tan46.27}{\tan46.27 - \tan40.3}\approx329.87$

Then h = (x + 100) * tan(46.27).
• Jan 16th 2013, 03:21 AM
Imonars
Re: Pyramid
h = 449.3 :)
Thank you we got it!!
• Jan 16th 2013, 03:30 AM
emakarov
Re: Pyramid
Strictly speaking, if we round the height to one decimal digit, it should be 449.4 because the following digit is 6.
• Jan 16th 2013, 06:01 AM
Soroban
Re: Pyramid
Hello, Imonars!

I would set it up like this:

Code:

                                  o C                               * * |                           *  *  |                       *    *    |                   *      *      |h               *        *        |           * β        * α        |     B o - - - - - - o - - - - - - o D       :    100    A      x      :
A man stands at $\displaystyle A$ and sights the top of the pyramid $\displaystyle C$.
The angle of elevation is $\displaystyle \alpha = 46.27^o.$

He moves 100 feet away to point $\displaystyle B\!:\;AB = 100.$
The angle of elevation is $\displaystyle \beta = 40.3^o.$

The height of the pyramid is: $\displaystyle h = CD.$
Let $\displaystyle x = AD.$

In $\displaystyle \Delta CDA\!:\;\tan\alpha \,=\, \frac{h}{x} \quad\Rightarrow\quad x \,=\, \frac{h}{\tan\alpha}$ .[1]

In $\displaystyle \Delta CDB\!:\;\tan\beta \,=\, \frac{h}{x+100} \quad\Rightarrow\quad x \,=\,\frac{h}{\tan\beta} - 100$ .[2]

Equate [2] and [1]: .$\displaystyle \frac{h}{\tan\beta} - 100 \;=\;\frac{h}{\tan\alpha}$

Multipy by $\displaystyle \tan\alpha\tan\beta\!:\;h\tan\alpha - 100\tan\alpha\tan\beta \:=\:h\tan\beta$

. . $\displaystyle h\tan\alpha - h\tan\beta \:=\:100\tan\alpha\tan\beta$

. . $\displaystyle h(\tan\alpha - \tan\beta) \:=\:100\tan\alpha\tan\beta$

. . . . . . . . . . . . $\displaystyle h \:=\:\frac{100\tan\alpha\tan\beta}{\tan\alpha - \tan\beta}$

Therefore: .$\displaystyle h \;=\;\frac{100\tan46.27^o\tan40.3^o}{\tan46.27^o - \tan40.3^o} \;=\; 449.364461\hdots$
• Jan 16th 2013, 06:05 AM
Imonars
Re: Pyramid
That's how we got it, we handed in the paper 2 hours ago, the work is the same :) Thanks a alot