Hello, euclid2!

I like skeeter's solution . . . Here's my version.

Vivian and Bobby are 250 m apart and are facing each other.

Each one is looking up at a hot air balloon.

The angle of elevation from Vivian to the balloon is 75°

and the angle of elevation from Bobby to the balloon is 50°.

Determine the height of the balloon, to one decimal place. Code:

H
*
*: *
* 55° *
* : * v
* : *
* :y *
* : *
* 75° : 50° *
V * * * * * * * B
h = 250

We have: .$\displaystyle VB = h = 250$

Since $\displaystyle \angle V = 75^o,\;\angle B = 50^o$, we have: .$\displaystyle \angle H = 55^o$

And we want $\displaystyle y.$

Law of Sines: .$\displaystyle \frac{v}{\sin V} = \frac{h}{\sin H} \quad\Rightarrow\quad v \:=\:\frac{250\sin75^o}{\sin55^o} \:\approx\:294.8$

We have: .$\displaystyle \sin B = \frac{y}{v}\quad\Rightarrow\quad y \:=\:v\!\cdot\!\sin B \:=\:294.8\sin50^o \;=\;\boxed{225.8\text{ m}}$