Hello, Damon!

From a window 4.2m above the ground,

the angle of depression ot the foot of the building across the road is 24°

the angel of elevation of the top of the same building is 34°.

Determine to the nearest cm the width of the road and the height of the building.

Here's the diagram for ticbol's solution. Code:

* C
* |
* |
* |
* |y
* |
* |
* 34° w |
A * - - - - - - - - - - - + D
| * 24° |
| * |
4.2 | * | 4.2
| * |
| * |
B +-----------------------* E
: - - - - w - - - - - :

The window is at $\displaystyle A$, 4.2 m above the ground.

. . $\displaystyle AB\,=\,DE\,=\,4.2$

Let $\displaystyle y \,=\,CD$ and $\displaystyle w \,=\,BE\,=\,AD.$

In right triangle $\displaystyle ADE\!:$

. . $\displaystyle \tan24^o \:=\:\frac{4.2}{w}\quad\Rightarrow\quad w \:=\:\frac{4.2}{\tan24^o}\quad\Rightarrow\quad \boxed{w \:\approx\:9.43\text{ m}}$

In right triangle $\displaystyle CDE:$

. . $\displaystyle \tan34^o \:=\:\frac{y}{w}\quad\Rightarrow\quad y \:=\:9.43\tan34^o\quad\Rightarrow\quad y \:\approx\:6.36\text{ cm}$

The height of the building is about: .$\displaystyle 6.36 + 4.20 \:=\:\boxed{10.56\text{ m}}$