To find out how long it will take for Francesca and Mercedes to arrive at the store, we can analyze their paths based on the coordinates provided.
From the information given:
- Francesca's path, \( F(m) \), starts at (0,10) and goes to (16,22).
- Mercedes' path, \( M(m) \), starts at (0,0) and goes to (14,28).
- They intersect at the point (8,16).
To determine how long it takes for each of them to reach the intersection point (8,16), we can look at the coordinates of their respective paths:
- Francesca (F):
- Start: (0,10)
- Intersection point: (8,16)
To find the time taken, we can compute the distance they travel along their respective paths to the intersection point.
Using the coordinates (0,10) to (8,16), we can find the distance:
- Change in x (Δx) = 8 - 0 = 8
- Change in y (Δy) = 16 - 10 = 6
- Distance (d) can be calculated using the Pythagorean theorem: \[ d_F = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(8)^2 + (6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
- Mercedes (M):
- Start: (0,0)
- Intersection point: (8,16)
For Mercedes, the distance from (0,0) to (8,16) is:
- Change in x (Δx) = 8 - 0 = 8
- Change in y (Δy) = 16 - 0 = 16
- The distance is: \[ d_M = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(8)^2 + (16)^2} = \sqrt{64 + 256} = \sqrt{320} = 8\sqrt{5} \approx 17.89 \]
Next, we need the velocities (or rates) for both Francesca and Mercedes to determine the time taken to reach the intersection.
Assuming the velocities are constant:
- Francesca moves from (0,10) to (8,16), which takes about 10 units of distance.
- Mercedes moves from (0,0) to (8,16), which takes about 17.89 units of distance.
To calculate the time taken for each to reach the meeting point, we need their speeds, which we currently don't have. However, if we presume they are in a consistent ratio based on how much distance they’ve covered in a given time frame until the intersection, we can use the intersection point to estimate times.
Given the points, we already know:
- They meet after traveling their respective distances at their own rates.
Thus, the time for each can be assumed in common time units:
- At the meeting point, both have arrived.
However, you will need a specified speed or timeframe for either Francesca or Mercedes to compute exact times in hours or minutes. If both take 1 time unit to each travel their respective distance, they would arrive at the same time at the intersection point at \( t=1 \), where:
- Francesca has traveled 10 units of distance
- Mercedes has traveled 17.89 units of distance
Without exact rates, we cannot provide a numeric time in units (like minutes or seconds). So under typical conditions where rates are not provided, they simply arrive at the intersection at the same time.