Two students, each riding bicycles, start from the same apartment building and ride to the same building on campus, but each takes a different route. The first student rides 1300 m due east and then turns due north and travels another 1430 m before arriving at the destination. The second student heads due north for 1930 m and then turns and heads directly toward the destination. (a) At the turning point, how far is the second student from the destination? m
(b) During the last leg of the trip, what direction (measured relative to due east) must the second student head? (Give your answer as a positive number from 0 to 180 degrees, either north or south of due east.) degrees south of east

a) d= 1393 m

b) θ= 21º S of E.

Explanation:

a)

Since the second student goes due north, and the first student goes due east along 1300m till he turns directly northward, we conclude that when the second student turns, he is at 1300m west from the destination.Since he rode 1930 m due north, we can conclude also that the second student is 500 m past the destination in the north direction.So we can find the distance from the destination at the turning point, using the Pythagorean Theorem, taking the right triangle defined by the 1300 m  segment due east, the 500 m segment due south, and which hypotenuse is the distance straight to the destination, as follows:

       

b)

Taking the same right triangle than in (a), we can find the angle that makes the vector along the direction taken by the second student with the due east, applying the definition of tangent of an angle, as follows:

       

       ⇒ θ= tg⁻¹ (0.385) = 21º S of E.