Calculate the height of a building This question I really need help with, I simply do not know where to start! Anyone can help, all I can offer is supreme thanks. Please include method. I don't want simple answers which don't help me learn
 A: Let the length of legs of the small right triangle be $h,x$, then use trigonometry to get two relations including these unknowns
$$\frac{h}{20+x}=\tan 23 \\ \frac{h}{x}=\tan 39$$
A: Let $h$ be the height of building and $d$ be the distance from $Y$ to the bottom of building. Therefore, we have the following relations:
$$
\tan 39^\circ=\frac hd\quad\Rightarrow\quad d=\frac h{\tan 39^\circ}\tag1
$$
and
$$
\tan 23^\circ=\frac h{d+20}\quad\Rightarrow\quad d+20=\frac h{\tan 23^\circ}.\tag2
$$
Substitute $d$ in $(1)$ to $(2)$ yield
$$
\begin{align}
\frac h{\tan 39^\circ}+20&=\frac h{\tan 23^\circ}\\
20&=\frac h{\tan 23^\circ}-\frac h{\tan 39^\circ}\\
20&=h\left(\frac 1{\tan 23^\circ}-\frac 1{\tan 39^\circ}\right)\\
20&=h\left(\frac {\tan 39^\circ-{\tan 23^\circ}}{\tan 23^\circ{\tan 39^\circ}}\right)\\
\Large\color{blue}h&=\Large\color{blue}{\frac{20\tan 23^\circ{\tan 39^\circ}}{\tan 39^\circ-{\tan 23^\circ}}}.
\end{align}
$$
A: Let the point falling down from Z be T and the distance from Y to T be m
Apply tanQ in triangle XZT and triangle YZT which will give 
tan 23 = x/(20+m) & tan 39 = x/m 
Solve thes two linear equations you will get your answer
A: $$\begin{align}
&\tan 39^\circ=h/w \implies w=\frac{h}{\tan 39^\circ} \; \tag{*}\\
&\tan 23^\circ=\frac{h}{(20+w)} \implies 20\tan23^\circ+w\tan23^\circ=h \; \tag{#}\\
&\text{Hence from $*$ and $\#$,}\;\; 20\tan23^\circ+\tan23^\circ\times\left({{h}\over{\tan39^\circ}}\right)=h\\
&\rightarrow h=(20\tan23^\circ)\cdot \left(1-\frac{\tan23^\circ}{\tan39^\circ}\right)=4.039..\approx 4.04 \;\rm{m}
\end{align}$$
A: The way I solve problems like these (and word problems in general) is


*

*identify what you want to solve for (in this case the height $h$)

*identify all the relationships that you can. "This distance is 20" is not a relationship; it needs to be part of an equation (or inequality).

*don't be afraid of creating more than 1 variable. You can eliminate the ones you don't want by doing substitutions later on. as long as you have as many relationships as variables, you'll be fine.

