A family is traveling due west on a road that is passes a famous landmark. At a given time the bearing to the landmark is N 62° W, and after the family travels 5 miles farther the bearing is N 38° W. What is the closest the family will come to the landmark while on the road?

Question

A family is traveling due west on a road that is passes a famous landmark.  At a given time the bearing to the landmark is N 62° W, and after the family travels 5 miles farther the bearing is N 38° W.  What is the closest the family will come to the landmark while on the road?

anonymous · Answer

@DatChinookGuy

anonymous · Answer

I'm not sure why so many questions include "bearing" so that's what's really messing me up because I don't know what that is.

anonymous · Answer

Also, I have to create a drawing for each question, so if you have time to help me with one for this question that'd be great :)

anonymous · Answer

The bearing of N 62 degrees W tell you to look North and then rotate 62o toward the West as in the diagram. In the diagram R is the closest point on the road to the landmark, so angle LRP is a right angle. 

In triangle LQP you know two of the angles, 27o and 38o + 90o = 128o so you can find the remaining angle PLQ. Use the Law of Sines to find the length of LQ and then you can find the length of LR since sin(52o) = |LR|/|LQ| 
You probably need to draw a picture of the situation. The bearing in each case is the hypotenuse of a right triangle. The road westward toward the point closest to the landmark is one side. The perpendicular from the landmark to the road is the other side. 

 At the first time mentioned, the angle between the road and the bearing to the landmark (that is, the hypotenuse) is 
 90 - 62 = 38 degrees 

 At the second time, it's 
 90 - 38 = 62 degrees 

 Let's let the distance to the landmark be x. That's the length of the side opposite each of these angles. 

 Let's let the distances to the point on the road closest to the landmark (that is, to the right angle of the triangles) be 
 d at the second observation, and 
 d+5 at the first one. 

 The ratio between the family's distance from that closest point, and the landmark's distance from the road, is the tangent of the angle of the right triangle at the landmark. That is, 
 d/x = tan 38 and 
 (d+5)/x = tan 62 

 Therefore 
 (d+5)/x - d/x = tan 62 - tan 38 
 d + 5 - d = x (tan 62 - tan 38) 
 x = 5 / (tan 62 - tan 38) = about 4.55 miles 
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