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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 [tex] \theta_{1}=\angle BAC[/tex] and [tex] \theta_{2}=\angle ABD[/tex] as follows,
For [tex] \theta_{1}=\angle BAC[/tex]
[tex] \angle XAC=\angle XAB+\angle BAC[/tex]
[tex] \therefore 80\degree=65\degree+\angle BAC[/tex]
[tex] \therefore \angle BAC=15\degree[/tex]
For [tex] \theta_{2}=\angle ABD[/tex]
By using alternate angle property,
[tex] \ angle XAB =\angle ABD=65\degree[/tex]
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.
[tex]\angle BAC+\angle ABC+\angle BCA=180\degree[/tex]
[tex] \therefore15\degree +135\degree +\angle BCA=180\degree[/tex]
[tex] \therefore \angle BCA=30\degree[/tex]
Refer attachment 2.
Sine rule for the [tex] \Delta ABC [/tex]can be applied as follows,
[tex]\ddfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}[/tex]
Substituting the values,
[tex]\dfrac{\sin 15}{a}=\dfrac{\sin 135}{b}=\dfrac{\sin 30}{30}[/tex]
Now distance AC and BC can be calculated as follows,
For distance AC,
[tex]\dfrac{\sin 135}{b}=\dfrac{\sin 30}{30}[/tex]
Cross multiplying,
[tex]30\times\sin 135=b\times\sin 30[/tex]
[tex]\dfrac{30\times \sin 135}{\sin 30}=b[/tex]
[tex]b=42.43\:km[/tex]
The distance of fire from Pine Knob fire tower is 42.43 km
For distance BC,
[tex]\dfrac{\sin 15}{a}=\dfrac{\sin 135}{42.43}[/tex]
Cross multiplying,
[tex]42.43\times\sin 15=a\times\sin 135[/tex]
[tex]\dfrac{42.43\times \sin 15}{\sin 135}=a[/tex]
[tex]a=15.53\:km[/tex]
The distance of fire from Pine Knob fire tower is 15.53 km
The distance of the fire from each tower will be 42.43km and 15.53 km respectively.
How to calculate the distance?
Firstly, we need to get the value of angle BAC. This will be:
BAC = 80° - 65° = 15°
By using sine rule,
Sin 15/a = sin 135/b = sin 30/30.
b = (30 × sin 135) / sin 30
b = 42.43 km
Also, the value of a will be:
= (42.43 × sin 15) / sin 135
= 15.53 km
Therefore, the distance of the fire from each tower will be 42.43km and 15.53 km respectively.
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