Hello, sinjid9!

The angle of elevation to a building is 30°.

From a point 20 m closer to the building, the angle of elevation is 40°.

Find the height of the building. Include a diagram in your solution. Code:

A o
| * *
| * *
h | * *
| * *
| 40° * 30° *
B o - - - - - - - - o - - - - - - - - o
x D 20 C

The building is: $\displaystyle h \,=\,AB.$

$\displaystyle \angle ACB = 30^o,\;\angle ADB \,=\,40^o,\;DC \,=\,20$

Let $\displaystyle x \,=\,BD.$

In right triangle $\displaystyle ABC\!:\;\;\tan30 \:=\:\frac{h}{x+20} \quad\Rightarrow\quad x \:=\:\frac{h-20\tan30}{\tan30} $ .[1]

In right triangle $\displaystyle ABD\!:\;\;\tan40 \:=\:\frac{h}{x} \quad\Rightarrow\quad x \:=\:\frac{h}{\tan40}$ .[2]

Equate [1] and [2]: .$\displaystyle \frac{h-20\tan30}{\tan30} \;=\;\frac{h}{\tan40} \quad\Rightarrow\quad h\tan40 - 20\tan30\tan40 \;=\;h\tan30 $

. . . . . . . .$\displaystyle h\tan40 - h\tan30 \:=\:20\tan30\tan40 \quad\Rightarrow\quad h(\tan40-\tan30) \:=\:20\tan30\tan40$

Therefore: .$\displaystyle h \;=\;\frac{20\tan30\tan40}{\tan40-\tan30} \;=\;37.01666314 \;\approx\;37\text{ m}$