Answer :
To solve this problem, let's analyze and verify the distances and displacements of each runner based on their given speed, velocity, and time.
### Step 1: Convert Time to Hours
Time is given in minutes, so we need to convert it to hours to use with the given speed (which is in mph).
- 1 hour = 60 minutes
So, convert the times:
- Garreth: 40 minutes = 40 / 60 = 2/3 hours
- Luigi: 35 minutes = 35 / 60 = 35/60 hours (7/12 hours)
- Elisa: 40 minutes = 40 / 60 = 2/3 hours
- Jasmine: 35 minutes = 35 / 60 = 35/60 hours (7/12 hours)
- Logan: 35 minutes = 35 / 60 = 35/60 hours (7/12 hours)
### Step 2: Calculate Distances Traveled
Distance [tex]\( d \)[/tex] is given by [tex]\( d = \text{speed} \times \text{time} \)[/tex]:
- Garreth: [tex]\( 6.2 \ \text{mph} \times \frac{2}{3} \ \text{hours} = 4.13 \ \text{miles} \)[/tex]
- Luigi: [tex]\( 5.0 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.92 \ \text{miles} \)[/tex]
- Elisa: [tex]\( 6.2 \ \text{mph} \times \frac{2}{3} \ \text{hours} = 4.13 \ \text{miles} \)[/tex]
- Jasmine: [tex]\( 5.0 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.92 \ \text{miles} \)[/tex]
- Logan: [tex]\( 8.6 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 5.02 \ \text{miles} \)[/tex]
### Step 3: Calculate Displacements
Displacement is given by [tex]\( \text{velocity} \times \text{time} \)[/tex]:
- Garreth: [tex]\( 5.5 \ \text{mph} \times \frac{2}{3} \ \text{hours} = 3.67 \ \text{miles \ west} \)[/tex]
- Luigi: [tex]\( 4.3 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.52 \ \text{miles \ east} \)[/tex]
- Elisa: [tex]\( 6.5 \ \text{mph} \times \frac{2}{3} \ \text{hours} = 4.33 \ \text{miles \ west} \)[/tex]
- Jasmine: [tex]\( 4.0 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.33 \ \text{miles \ south} \)[/tex]
- Logan: [tex]\( 4.0 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.33 \ \text{miles \ north} \)[/tex]
### Step 4: Verify Statements
1. Garreth and Elisa ran the same distance but not the same displacement, Luigi and Jasmine went the same displacement, and Logan ran faster than Jasmine.
- Garreth and Elisa had the same distance (4.13 miles) but different displacements (3.67 miles west and 4.33 miles west, respectively).
- Luigi and Jasmine did not have the same displacement (Luigi 2.52 miles east and Jasmine 2.33 miles south).
- Logan ran at a speed of 8.6 mph which is indeed faster than Jasmine's speed of 5.0 mph.
2. Garreth ran farther than Luigi, Elisa and Jasmine went the same distance, and Logan and Jasmine have the same displacement in the opposite direction.
- Garreth's distance (4.13 miles) is indeed farther than Luigi's distance (2.92 miles).
- Elisa and Jasmine do not have the same distance (Elisa 4.13 miles and Jasmine 2.92 miles).
- Logan and Jasmine both have a displacement of 2.33 miles but in opposite directions (north and south, respectively).
3. Garreth and Elisa ran together, Luigi and Jasmine went the same distance, and Logan and Jasmine have the same displacement in the opposite direction.
- Garreth and Elisa ran the same distance (4.13 miles), suggesting they ran together.
- Luigi and Jasmine did not have the same distance (Luigi 2.92 miles and Jasmine 2.92 miles).
- Logan and Jasmine indeed had the same displacement of 2.33 miles in opposite directions.
4. Garreth ran less than Logan, Elisa and Jasmine went the same displacement, and Luigi and Garreth ran together.
- Garreth's distance (4.13 miles) is less than Logan's (5.02 miles).
- Elisa and Jasmine have different displacements (4.33 miles west and 2.33 miles south, respectively).
- Luigi and Garreth did not run together as they had different distances.
Since the analysis shows multiple inconsistencies with each of the statements provided, no statement accurately describes the conditions for all runners.
Thus, the correct answer is that none of the given statements accurately describe the runners' distances and displacements.
### Step 1: Convert Time to Hours
Time is given in minutes, so we need to convert it to hours to use with the given speed (which is in mph).
- 1 hour = 60 minutes
So, convert the times:
- Garreth: 40 minutes = 40 / 60 = 2/3 hours
- Luigi: 35 minutes = 35 / 60 = 35/60 hours (7/12 hours)
- Elisa: 40 minutes = 40 / 60 = 2/3 hours
- Jasmine: 35 minutes = 35 / 60 = 35/60 hours (7/12 hours)
- Logan: 35 minutes = 35 / 60 = 35/60 hours (7/12 hours)
### Step 2: Calculate Distances Traveled
Distance [tex]\( d \)[/tex] is given by [tex]\( d = \text{speed} \times \text{time} \)[/tex]:
- Garreth: [tex]\( 6.2 \ \text{mph} \times \frac{2}{3} \ \text{hours} = 4.13 \ \text{miles} \)[/tex]
- Luigi: [tex]\( 5.0 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.92 \ \text{miles} \)[/tex]
- Elisa: [tex]\( 6.2 \ \text{mph} \times \frac{2}{3} \ \text{hours} = 4.13 \ \text{miles} \)[/tex]
- Jasmine: [tex]\( 5.0 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.92 \ \text{miles} \)[/tex]
- Logan: [tex]\( 8.6 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 5.02 \ \text{miles} \)[/tex]
### Step 3: Calculate Displacements
Displacement is given by [tex]\( \text{velocity} \times \text{time} \)[/tex]:
- Garreth: [tex]\( 5.5 \ \text{mph} \times \frac{2}{3} \ \text{hours} = 3.67 \ \text{miles \ west} \)[/tex]
- Luigi: [tex]\( 4.3 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.52 \ \text{miles \ east} \)[/tex]
- Elisa: [tex]\( 6.5 \ \text{mph} \times \frac{2}{3} \ \text{hours} = 4.33 \ \text{miles \ west} \)[/tex]
- Jasmine: [tex]\( 4.0 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.33 \ \text{miles \ south} \)[/tex]
- Logan: [tex]\( 4.0 \ \text{mph} \times \frac{7}{12} \ \text{hours} = 2.33 \ \text{miles \ north} \)[/tex]
### Step 4: Verify Statements
1. Garreth and Elisa ran the same distance but not the same displacement, Luigi and Jasmine went the same displacement, and Logan ran faster than Jasmine.
- Garreth and Elisa had the same distance (4.13 miles) but different displacements (3.67 miles west and 4.33 miles west, respectively).
- Luigi and Jasmine did not have the same displacement (Luigi 2.52 miles east and Jasmine 2.33 miles south).
- Logan ran at a speed of 8.6 mph which is indeed faster than Jasmine's speed of 5.0 mph.
2. Garreth ran farther than Luigi, Elisa and Jasmine went the same distance, and Logan and Jasmine have the same displacement in the opposite direction.
- Garreth's distance (4.13 miles) is indeed farther than Luigi's distance (2.92 miles).
- Elisa and Jasmine do not have the same distance (Elisa 4.13 miles and Jasmine 2.92 miles).
- Logan and Jasmine both have a displacement of 2.33 miles but in opposite directions (north and south, respectively).
3. Garreth and Elisa ran together, Luigi and Jasmine went the same distance, and Logan and Jasmine have the same displacement in the opposite direction.
- Garreth and Elisa ran the same distance (4.13 miles), suggesting they ran together.
- Luigi and Jasmine did not have the same distance (Luigi 2.92 miles and Jasmine 2.92 miles).
- Logan and Jasmine indeed had the same displacement of 2.33 miles in opposite directions.
4. Garreth ran less than Logan, Elisa and Jasmine went the same displacement, and Luigi and Garreth ran together.
- Garreth's distance (4.13 miles) is less than Logan's (5.02 miles).
- Elisa and Jasmine have different displacements (4.33 miles west and 2.33 miles south, respectively).
- Luigi and Garreth did not run together as they had different distances.
Since the analysis shows multiple inconsistencies with each of the statements provided, no statement accurately describes the conditions for all runners.
Thus, the correct answer is that none of the given statements accurately describe the runners' distances and displacements.
