Question

A fire in a building B is reported on telephone to two fire stations P and Q, 20 km a part from each other on a straight road. P observes that the fire is at an angle of 60° to the road and Q observes that it is at an angle of 45° to the road. Which station should send its team and how much will this team have to travel?

Answer 1

Let AB be the building of height h. P Observes that the fire is at an angle of 60° to the road and Q observes that the fire is at an angle of 45° to the road.

Let QA = x, AP = y. And , , given PQ = 20.

Here, clearly.

So stationis near to the building. Hence station P must send its team.

We sketch the following figure

So we use trigonometric ratios.

In ΔPAB

$$\begin{gathered} \tan P=\frac{AB}{AP} \\ \Rightarrow \tan 60°=\frac{h}{y} \\ \Rightarrow \sqrt{3}=\frac{h}{y} \\ \Rightarrow h=\sqrt{3}y \end{gathered}$$

Again in ΔQAB,

Now,

$$\begin{gathered} x+y=20 \\ \Rightarrow h+y=20\left[\because x=h\right] \\ \Rightarrow \sqrt{3}y+y=20\left[\because h=\sqrt{3}y\right] \\ \Rightarrow y=\frac{20}{\left(\sqrt{3}+1\right)}=10\left(\sqrt{3}-1\right) \end{gathered}$$

Hence, the team from station P will have to travel $10\left(\sqrt{3}-1\right)$ km.