Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides 1490 m due east and then turns due north and travels another 1430 m before reaching the campground. The second cyclist starts out by heading due north for 2130 m and then turns and heads directly toward the campground.

Question

Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides 1490 m due east and then turns due north and travels another 1430 m before reaching the campground. The second cyclist starts out by heading due north for 2130 m and then turns and heads directly toward the campground.
(a) At the turning point, how far is the second cyclist from the campground?
(blank) m

(b) What direction (measured relative to due east) must the second cyclist head during the last part of the trip?
(blank)° (blank = either north or south) of east

MathMate · Answer

Resolve the displacements into East (x) and North (y) components.
First find the coordinates of the campground relative to the starting point:
Distance angle x-component y-component
1490 0° 1490cos(0) 1490sin(0)
1430 90° 1430cos(90) 1430sin(90)
Total ---- 1490 1430
Therefore the campground is at (1490,1430) relative to the starting point.

At the turning point, the second cyclist is at (0,2130) relative to the starting point.
The distance from the campground is therefore the distance between the two points (0,2130) and (1490,1430).

The direction of P2(x2,y2) from P1(x1,y1) is atan2(y2-y1, x2-x1), taking into account of the signs of the values and therefore the quadrant.
In the case in point, the angle is
atan2(1430-2130,1490-0) = -25 degrees approximately, or 25 degrees south of east.