Two bicyclists, starting at the same place, are riding toward the same campground by different routes. One cyclist rides 1040 m due east and then turns due north and travels another 1540 m before reaching the campground. The second cyclist starts out by heading due north for 1860 m and then turns and heads directly toward the campground. (a) At the turning point, how far is the second cyclist from the campground? (b) What direction, measured as a positive angle relative to due east, must the second cyclist head during the last part of the trip?
To solve this problem, we can draw a diagram and use the Pythagorean theorem to find the distances and angles involved.
a) Let's start with the first cyclist. The cyclist rides east for 1040 m, then turns north and travels 1540 m. Drawing a right-angled triangle for this route, we can see that the distance traveled by the first cyclist can be found using the Pythagorean theorem:
Distance = √((1040)^2 + (1540)^2)
Using a calculator, we can find that the distance traveled by the first cyclist is approximately 1875.05 m.
b) Now let's move on to the second cyclist. The second cyclist starts by heading north for 1860 m. We need to find the distance between the turning point and the campground. Drawing another right-angled triangle for this route, we can again use the Pythagorean theorem:
Distance = √((1875.05)^2 - (1860)^2)
Using a calculator, we can find that the distance between the turning point and the campground is approximately 327.79 m.
c) To find the direction in which the second cyclist must head during the last part of the trip, we can use trigonometry.
Angle = inverse tangent (1860/1875.05)
Using a calculator, we can find that the angle is approximately 46.26 degrees.
So, the answer to part (a) is approximately 327.79 m, and the answer to part (b) is approximately 46.26 degrees (measured as a positive angle relative to due east).