Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides 1500 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 1870 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 relative to due east) must the second cyclist head during the last part of the trip?

To solve this problem, we can use the Pythagorean Theorem and trigonometry.

(a) To find the distance of the second cyclist from the campground at the turning point, we can use the Pythagorean Theorem. The distance traveled due north is 1870 m, and we need to find the remaining distance.

Using the Pythagorean Theorem, the distance from the turning point to the campground is:

x² = (1500 m)² + (1430 m)²
x² = 2250000 m² + 2044900 m²
x² = 4294900 m²

Taking the square root of both sides, we get:

x ≈ √(4294900 m²)
x ≈ 2073.81 m

Therefore, at the turning point, the second cyclist is approximately 2073.81 m away from the campground.

(b) To find the direction the second cyclist must head during the last part of the trip, we can use trigonometry. We can use the tangent function to find the angle.

Let θ be the angle:

tan(θ) = (1500 m) / (1430 m)
θ = tan^(-1)((1500 m) / (1430 m))
θ ≈ 46.37°

Therefore, the second cyclist must head in a direction approximately 46.37° relative to due east.