Hello, Vivin Spinach!

I don't agree with their answer . . .

Two spotlights, one blue and the other white, are placed 6.0m apart on a track

on the ceiling of a ballroom.

An observer on the floor sees that the angle of elevation is 45° to the blue spotlight

and 70° to the white one.

How high, to the nearest tenth of a metre, is the ceiling of the ballroom? Code:

W 6 B
C - - - - - * - - - - * -
70° * 110° * |
* * |
* * |
* * |
* * | h
*25°* |
* * |
* * |
** 45° |
O * - - - - - - - - - * A

The white light is at $\displaystyle W$, the blue light at $\displaystyle B.\;\;WB = 6$

The observer is at $\displaystyle O.\;\;\angle BOA = 45^o,\;\;\angle WOA = 70^o \quad\Rightarrow\quad \angle WOB = 25^o$

. . We also have: .$\displaystyle \angle CWO = 70^o \quad\Rightarrow\quad \angle BWO = 110^o$

Let $\displaystyle h = AB$

In $\displaystyle \Delta BWO$, use the Law of Sines:

. . $\displaystyle \frac{OB}{\sin110^o} \:=\:\frac{6}{\sin25^o} \quad\Rightarrow\quad OB \:=\:\frac{6\sin110^o}{\sin25^o} \:\approx\:13.34$

In $\displaystyle \Delta BAO\!:\;\;\sin45^o \:=\:\frac{h}{OB} \quad\Rightarrow\quad h \:=\:OB\sin45^o \:=\:13.34\sin45^o$

Therefore: .$\displaystyle OB \:\approx 9.4\text{ m}$