Question

The bearing from the Pine Knob fire tower to the Colt Station fire tower is N 65°E and the two towers are 30 kilometers apart. A fire spotted by rangers in each tower has a bearing of N 80° E from Pine Knob and S 70°E from Colt Station. Find the distance of the fire from each tower.

Answer 1

Answer:

The distance of fire from Pine Knob fire tower is 42.43 km

The distance of fire from Colt Station fire tower is 15.53 km

Step-by-step explanation:

Consider point A as Pine Knob fire tower and Point B as Colt Station fire tower. So the distance between both towers that is AB is 30 km.  

Now both fire tower  are located at a bearing of N65°E. That is at point A from north, Colt station fire tower is located at an angle of 65° towards east.

From ranger on Pine knob tower, fire spotted makes bearing of N80°E. That is at point A from north fire is located at an angle of 80° towards east.

Similarly, from ranger on Pine knob tower, fire spotted makes bearing of S70°E. That is at point B from south fire is located at an angle of 70° towards east.  

Now calculate $\theta_{1}=\angle BAC$ and $\theta_{2}=\angle ABD$ as follows,  

For $\theta_{1}=\angle BAC$

$\angle XAC=\angle XAB+\angle BAC$

$\therefore 80\degree=65\degree+\angle BAC$

$\therefore \angle BAC=15\degree$

For $\theta_{2}=\angle ABD$

By using alternate angle property,

$\ angle XAB =\angle ABD=65\degree$

Refer attachment 1.  

From diagram consider the triangle ABC. To find the third angle that is \angle BAC can be calculated by using angle sum property of triangle.  

$\angle BAC+\angle ABC+\angle BCA=180\degree$

$\therefore15\degree +135\degree +\angle BCA=180\degree$

$\therefore \angle BCA=30\degree$

Refer attachment 2.  

Sine rule for the $\Delta ABC$can be applied as follows,  

$\ddfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$

Substituting the values,

$\dfrac{\sin 15}{a}=\dfrac{\sin 135}{b}=\dfrac{\sin 30}{30}$

Now distance AC and BC can be calculated as follows,  

For distance AC,  

$\dfrac{\sin 135}{b}=\dfrac{\sin 30}{30}$

Cross multiplying,  

$30\times\sin 135=b\times\sin 30$

$\dfrac{30\times \sin 135}{\sin 30}=b$

$b=42.43\:km$

The distance of fire from Pine Knob fire tower is 42.43 km

For distance BC,  

$\dfrac{\sin 15}{a}=\dfrac{\sin 135}{42.43}$

Cross multiplying,  

$42.43\times\sin 15=a\times\sin 135$

$\dfrac{42.43\times \sin 15}{\sin 135}=a$

$a=15.53\:km$

The distance of fire from Pine Knob fire tower is 15.53 km