Two mountain bikers have been separated and are hoping to find each other. One biker reads on her GPS that she is located at 102.9 degrees west and 44.1 degrees north. She decides to ride towards the mountain peak, which on the map has the coordinates 102.5 degrees west and 44.3 degrees north. Her friend sees on his map that he is at the bend in the trail located at 102.5 degrees west and 44.3 degrees north. He decides to ride toward the waterfall with coordinates 102.9 degrees west and 44.5 degrees north. Will the two bikers cross paths? If so, at what coordinates? If not, why not?
To determine if the two mountain bikers will cross paths, we need to compare the coordinates at which they are currently located and the direction in which they are riding.
The first biker is currently located at 102.9 degrees west and 44.1 degrees north. She is riding towards the mountain peak located at 102.5 degrees west and 44.3 degrees north.
The second biker is currently located at 102.5 degrees west and 44.3 degrees north. He is riding towards the waterfall located at 102.9 degrees west and 44.5 degrees north.
If we plot these coordinates on a map, we can see that the two bikers are moving towards each other. The first biker is moving northwest, and the second biker is moving southeast.
Since their directions are roughly opposite, it is likely that the two bikers will eventually cross paths. However, the specific coordinates at which they will meet can be determined by calculating the midpoint between their current positions.
To find the midpoint, we can add the latitude coordinates and divide by 2, and add the longitude coordinates and divide by 2.
Latitude midpoint:
(44.1 + 44.3) / 2 = 44.2
Longitude midpoint:
(102.9 + 102.5) / 2 = 102.7
Therefore, the two bikers will cross paths at approximately 44.2 degrees north and 102.7 degrees west.
To determine if the two bikers will cross paths, we need to see if the paths they are riding on intersect. Let's analyze their coordinates and determine if their paths will converge.
The first biker's current location is at 102.9 degrees west and 44.1 degrees north. She decides to ride towards the mountain peak at 102.5 degrees west and 44.3 degrees north.
The second biker's current location is at 102.5 degrees west and 44.3 degrees north. He decides to ride toward the waterfall at 102.9 degrees west and 44.5 degrees north.
We can observe that both bikers are moving in the same general direction, from west to east. However, we need to determine if their paths will intersect at some point.
To analyze this, we need to compare the longitude (west-east position) and latitude (north-south position) coordinates separately.
Looking at the longitude coordinates:
- The first biker is at 102.9 degrees west and moving towards 102.5 degrees west.
- The second biker is at 102.5 degrees west and moving towards 102.9 degrees west.
The paths of the bikers are converging with respect to longitude. As they are moving towards each other, they will eventually cross paths along the east-west direction.
Next, let's examine the latitude coordinates:
- The first biker is at 44.1 degrees north and moving towards 44.3 degrees north.
- The second biker is at 44.3 degrees north and moving towards 44.5 degrees north.
The paths of the bikers are also converging in terms of latitude. As they are both moving towards higher latitudes, they will eventually cross paths along the north-south direction.
Based on the analysis of their coordinates, the two bikers will indeed cross paths. To determine the precise coordinates where they will meet, we can take the average of their respective longitude and latitude coordinates at that point.
Longitude: (102.9 + 102.5) / 2 = 102.7 degrees west
Latitude: (44.1 + 44.3) / 2 = 44.2 degrees north
Therefore, the two bikers will cross paths at approximately 102.7 degrees west and 44.2 degrees north.