Answer:
Shortest distance from the mountain is 3.17 miles.
Step-by-step explanation:
From the figure attached,
Let a mountain is located at point A.
Angle between the mountain and point B (∠B) = 53°
Angle between the mountain and point C (∠C) = 78°
Distance between these points = 3 miles
Since, m∠A + m∠B + m∠C = 180°
m∠A + 53° + 78° = 180°
m∠A = 180°- 131° = 49°
By applying sine rule in triangle ABC,
[tex]\frac{\text{sin}(49)}{BC}=\frac{\text{sin}(53)}{AC}= \frac{\text{sin}(78)}{AB}[/tex]
[tex]\frac{\text{sin}(49)}{3}=\frac{\text{sin}(53)}{AC}= \frac{\text{sin}(78)}{AB}[/tex]
[tex]\frac{\text{sin}(49)}{3}=\frac{\text{sin}(53)}{AC}[/tex]
AC = [tex]\frac{3\text{sin}(53)}{\text{sin}(49)}[/tex]
AC = 3.17 miles
[tex]\frac{\text{sin}(49)}{3}=\frac{\text{sin}(78)}{AB}[/tex]
AB = [tex]\frac{3\text{sin}(78)}{\text{sin}(49)}[/tex]
AB = 3.89 miles
Therefore, shortest distance from the mountain is 3.17 miles.
