1. ## Tower Question

The 3d triangles are getting me!!

For the first part by considering appropriate isosceles triangles and right-angled triangles I got $\displaystyle h = a \sin \alpha \sec \theta$ is that correct?

I'm also lost on the next part as well.

Bobak

2. Here's a plan of the view on the ground:

$\displaystyle \setlength{\unitlength}{3.5mm} \begin{picture}(30,18) \put(2,0){\line(0,1){15}} \put(2,0){\line(5,3){19.5}} \put(2,15){\line(5,-3){12.6}} \put(2,15){\line(6,-1){19.5}} \put(2,15){\line(3,-5){6.6}} \put(8,4.8){\line(5,3){1.1}} \put(9.5,4.5){\line(-3,5){0.62}} \put(0.5,0){$O$} \put(0.5,14.5){$T'$} \multiput(6,1.4)(6,3.6){3}{$a$} \put(2.5,1){$\theta$} \put(0.5,7){$x$} \put(14.5,6){$P$} \put(21.5,10.5){$Q$} \end{picture}$

Let $\displaystyle OT' = x$ (T' denotes the base of the tower, so I'm using x to denote the horizontal distance from O to the base of the tower, not the inclined distance to the top of the tower). Then $\displaystyle h/x = \tan\alpha$. From the picture, $\displaystyle a/x = \cos\theta$. Therefore $\displaystyle \frac ah = \frac{a/x}{h/x} = \frac{\cos\theta}{\tan\alpha}$, and so $\displaystyle h = a\tan\alpha\sec\theta$.

Next, $\displaystyle \frac h{QT'} = \tan\beta$, so $\displaystyle QT'\,^2 = h^2\cot^2\beta = a^2\tan^2\alpha\cot^2\beta\sec^2\theta$. But by Pythagoras (see the above diagram), $\displaystyle QT'\,^2 = a^2(4+\tan^2\theta).$ Thus $\displaystyle \tan^2\alpha\cot^2\beta\sec^2\theta = 4+\tan^2\theta$, and I guess you can take it from there to get the formula for tan^2(θ). Once you have that, the remaining two parts of the question are easy.

3. Thank you very much Opalg I understand the error in taking the inclined lengths.

just to check

c) use $\displaystyle tan^2 \theta > 0$

d) $\displaystyle \sqrt 2 : \sqrt 5$

Bobak

Off topic: how did you produce the latex drawing ?

4. Originally Posted by bobak

Off topic: how did you produce the latex drawing ?
They're called PStricks, and it's pretty complicated to dominate it but not impossible.

$$\setlength{\unitlength}{3.5mm} \begin{picture}(30,18) \put(2,0){\line(0,1){15}} \put(2,0){\line(5,3){19.5}} \put(2,15){\line(5,-3){12.6}} \put(2,15){\line(6,-1){19.5}} \put(2,15){\line(3,-5){6.6}} \put(8,4.8){\line(5,3){1.1}} \put(9.5,4.5){\line(-3,5){0.62}} \put(0.5,0){O} \put(0.5,14.5){T'} \multiput(6,1.4)(6,3.6){3}{a} \put(2.5,1){\theta} \put(0.5,7){x} \put(14.5,6){P} \put(21.5,10.5){Q} \end{picture}$$