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

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.

(a) To find the distance of the second cyclist from the campground at the turning point, we need to calculate the horizontal and vertical distances traveled by the first cyclist.

The first cyclist traveled 1490 m east and then turned north, traveling another 1430 m. This means they traveled a total horizontal distance of 1490 m and a vertical distance of 1430 m.

Since the second cyclist started by heading due north for 2130 m, their vertical distance is already 2130 m.

To find the distance between the second cyclist and the campground at the turning point, we can subtract the vertical distances traveled by both cyclists:

Distance = Vertical distance of second cyclist - Vertical distance of first cyclist
Distance = 2130 m - 1430 m
Distance = 700 m

Therefore, the second cyclist is 700 m away from the campground at the turning point.

(b) To find the direction the second cyclist must head during the last part of the trip, we need to determine whether they should go north or south of east.

Since the first cyclist turned north, it means the turning point is either north or south of east, depending on the direction of the second cyclist.

To determine this, we need to compare the horizontal distances traveled by both cyclists:

Horizontal distance of first cyclist = 1490 m
Horizontal distance of second cyclist = 0 m (since they start heading due north)

Since the second cyclist has not traveled any horizontal distance yet, they need to head east in order to reach the campground.

Therefore, the second cyclist must head east, which means the direction relative to due east is 0° south of east.

To solve this problem, we can use the Pythagorean theorem and trigonometric functions to find the distances and angles involved.

(a) Let's find the distance from the turning point to the campground for the second cyclist. We can create a right triangle with the distance traveled due north (2130 m) as the vertical leg and the distance from the turning point to the campground as the hypotenuse. Using the Pythagorean theorem, we have:

distance^2 = (distance traveled due north)^2 + (distance traveled due east)^2

Let's denote the distance from the turning point to the campground as 'x'.

x^2 = 2130^2 + (1490 + 1430)^2
x^2 = 2130^2 + 2920^2
x^2 = 4,536,900 + 8,534,400
x^2 = 13,071,300

Taking the square root of both sides, we find:

x ≈ √(13,071,300)
x ≈ 3612.69 m

Therefore, at the turning point, the second cyclist is approximately 3612.69 m away from the campground.

(b) To find the direction the second cyclist must head during the last part of the trip, we need to determine the angle at the turning point (relative to due east). We can use trigonometric ratios to find this angle.

Let's denote the angle at the turning point as 'θ'.

Using the tangent function, we have:

tan(θ) = (distance traveled due east) / (distance traveled due north)

tan(θ) = (1490 + 1430) / 2130
tan(θ) = 2920 / 2130
tan(θ) ≈ 1.37

To find the angle, we can use the inverse tangent function:

θ ≈ tan^(-1)(1.37)
θ ≈ 52.54°

Therefore, the second cyclist must head approximately 52.54° north of east during the last part of the trip.