Question

A fire in a building $B$ is reported on telephone to two fire stations $P$ and $Q$, $30km$ apart from each other on a straight road. $P$ observes that the fire is at an angle of ${60}^o$ to the road and $Q$ observes that it is at an angle of ${45}^o$ to the road. Which station should send its team and how much will this team have to travel?

Answer 1

Consider the given data.

The distance between $P$ and $Q$ is $30km$

In triangle $BPM$,

$\tan {60}^{\circ }=\frac{h}{x}$

$\sqrt{3}x=h$

In triangle BQM,

$\tan {45}^{\circ }=\frac{h}{30-x}$

$h=30-x$

$\sqrt{3}x=30-x$

$(\sqrt{3}+1)x=30$

$x=\frac{30}{\sqrt{3}+1}$

Therefore,

$x=10.98$

$x\approx 11km$

Therefore, the distance between $P$ and building is $11km.$

And the distance between $Q$ and building is $30-11=19km$.

So, station $P$ should send its team because it is nearest to the building.