Question

A fire in a building ‘B’ is reported on telephone in two fire stations P an Q, 20

km apart 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 distance will this team has

to travel? What value is depicted from the problem?

Answer 1

Hi there!

Let AB be the building which is caught on a fire.

Given, distance between two fire stations PQ = 10 km

Let Distance between building B and fire station Q be ‘x’ km
Hence, PB = (10 – x) km

In right ΔABQ, θ = 45°

$tan \: \alpha = \frac{AB}{BQ}$

$tan \: 45 \: = \frac{AB}{x}$

$1 \: = \: \frac{AB}{x}$

Therefore AB = x km

Now consider, ΔABQ

$tan \: 60 \: = \: \frac{AB}{BP}$

$\sqrt{3 \: } \: = \: \frac{x}{10 - x}$

10√3 - √3x = x

10√3 = √3x + x = x(√3 + 1)

$x \: = \: \frac{10 \sqrt{3} }{ \sqrt{3 } + 1 }$

$x \: = \: \frac{10 \sqrt{3} }{ \sqrt{3} + 1 } \: \times \: \frac{ \sqrt{3} - 1 }{ \sqrt{3} - 1 }$

x = 5√3(√3 - 1)

x = 5 × 1.732 × 0.732 = 6.34 km (approx.)

That is fire station Q is at a distance of 6.34 km from the building and fire station P is at a distance 3.66 km.
Hence, fire station P has to send his team first and it has to travel a distance of 3.66 km

Values depicted by station :
Keen Observation (They knew which fire team will arrive first)

[ Thank you! for asking the question. ]
Hope it helps!