Two bicyclists, starting at the same place, are riding toward the same campground by different routes. One cyclist rides 1120 m due east and then turns due north and travels another 1550 m before reaching the campground. The second cyclist starts out by heading due north for 1800 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?
Two bicyclists, starting at the same place, are riding toward the same campground by two different routes. One cyclist rides 1230 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 1830 m and then turns and heads directly toward the campground. In 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 concept of vectors and right triangles. Let's break down the problem into steps:
(a) To find the distance between the turning point and the campground for the second cyclist, we need to calculate the distance traveled by the first cyclist and subtract it from the total distance traveled by the second cyclist.
1. Calculate the distance traveled by the first cyclist:
- The first cyclist rides 1120 m due east.
- Then, the cyclist turns due north and travels 1550 m.
- This forms a right triangle with the eastward distance as the base and the northward distance as the height.
- Using the Pythagorean theorem (a^2 + b^2 = c^2), we can calculate the distance traveled by the first cyclist (c):
c = sqrt((1120^2) + (1550^2))
2. Calculate the distance traveled by the second cyclist:
- The second cyclist starts by heading due north for 1800 m.
- At the turning point, the second cyclist is perpendicular to the first cyclist's route.
- This makes the distance between the turning point and the campground the height of the right triangle.
- Using the Pythagorean theorem (a^2 + b^2 = c^2), we can calculate the distance traveled by the second cyclist (b):
b = sqrt((1800^2) - (c^2))
3. Calculate the distance between the turning point and the campground for the second cyclist:
- The total distance traveled by the second cyclist is equal to the distance traveled in step 2.
- The distance between the turning point and the campground is b.
(b) To find the direction the second cyclist must head during the last part of the trip, we need to calculate the angle between the eastward direction and the direction from the turning point to the campground.
1. Calculate the angle using trigonometry:
- The angle we need to find is the angle opposite to the eastward direction in the right triangle formed by the eastward distance, northward distance, and the total distance traveled.
- We can use the inverse tangent function (arctan) to calculate the angle:
angle = arctan(northward distance / eastward distance)
2. Convert the angle to positive direction relative to due east:
- The arctan function gives an angle between -90 degrees and 90 degrees.
- We need to convert it to a positive angle relative to due east.
- If the angle is negative, we can add 360 degrees to it to get the positive angle.
By following these steps, we can find the answers to both parts of the problem.