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Sagot :
To solve this problem, we'll break it down step by step and then provide a detailed explanation to find the average speed and average velocity of the cab.
### Given Data:
1. Distance from Delhi to Agra: [tex]\(240 \text{ km}\)[/tex]
2. Time taken to travel from Delhi to Agra in the morning: [tex]\(45 \text{ hours}\)[/tex]
3. Time taken to return from Agra to Delhi during the night: [tex]\(3.5 \text{ hours}\)[/tex]
### Step-by-Step Solution:
#### 1. Calculate the Total Distance:
Since the cab travels from Delhi to Agra and back from Agra to Delhi, the total distance covered is the sum of both the trips.
[tex]\[ \text{Total Distance} = \text{Distance from Delhi to Agra} + \text{Distance from Agra to Delhi} = 240 \text{ km} + 240 \text{ km} = 480 \text{ km} \][/tex]
#### 2. Calculate the Total Time:
The total time for the round trip is the sum of the time taken for both legs of the journey.
[tex]\[ \text{Total Time} = \text{Time taken from Delhi to Agra} + \text{Time taken from Agra to Delhi} = 45 \text{ hours} + 3.5 \text{ hours} = 48.5 \text{ hours} \][/tex]
#### 3. Calculate the Average Speed:
The average speed is the total distance covered divided by the total time taken.
[tex]\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{480 \text{ km}}{48.5 \text{ hours}} \approx 9.897 \text{ km/h} \][/tex]
#### 4. Calculate the Average Velocity:
Average velocity is defined as the total displacement divided by the total time taken. Displacement is the straight-line distance between the initial and final positions.
Since the cab returns to the original starting point (Delhi), the total displacement is zero.
[tex]\[ \text{Total Displacement} = \text{Final Position} - \text{Initial Position} = 0 \text{ km} \][/tex]
[tex]\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{0 \text{ km}}{48.5 \text{ hours}} = 0 \text{ km/h} \][/tex]
### Conclusion:
- The total distance covered during the round trip is [tex]\(480 \text{ km}\)[/tex].
- The total time taken for the round trip is [tex]\(48.5 \text{ hours}\)[/tex].
- The average speed of the cab is approximately [tex]\(9.897 \text{ km/h}\)[/tex].
- The average velocity of the cab is [tex]\(0 \text{ km/h}\)[/tex].
These results have been derived from the given data and provide a complete and precise solution to the problem.
### Given Data:
1. Distance from Delhi to Agra: [tex]\(240 \text{ km}\)[/tex]
2. Time taken to travel from Delhi to Agra in the morning: [tex]\(45 \text{ hours}\)[/tex]
3. Time taken to return from Agra to Delhi during the night: [tex]\(3.5 \text{ hours}\)[/tex]
### Step-by-Step Solution:
#### 1. Calculate the Total Distance:
Since the cab travels from Delhi to Agra and back from Agra to Delhi, the total distance covered is the sum of both the trips.
[tex]\[ \text{Total Distance} = \text{Distance from Delhi to Agra} + \text{Distance from Agra to Delhi} = 240 \text{ km} + 240 \text{ km} = 480 \text{ km} \][/tex]
#### 2. Calculate the Total Time:
The total time for the round trip is the sum of the time taken for both legs of the journey.
[tex]\[ \text{Total Time} = \text{Time taken from Delhi to Agra} + \text{Time taken from Agra to Delhi} = 45 \text{ hours} + 3.5 \text{ hours} = 48.5 \text{ hours} \][/tex]
#### 3. Calculate the Average Speed:
The average speed is the total distance covered divided by the total time taken.
[tex]\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{480 \text{ km}}{48.5 \text{ hours}} \approx 9.897 \text{ km/h} \][/tex]
#### 4. Calculate the Average Velocity:
Average velocity is defined as the total displacement divided by the total time taken. Displacement is the straight-line distance between the initial and final positions.
Since the cab returns to the original starting point (Delhi), the total displacement is zero.
[tex]\[ \text{Total Displacement} = \text{Final Position} - \text{Initial Position} = 0 \text{ km} \][/tex]
[tex]\[ \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} = \frac{0 \text{ km}}{48.5 \text{ hours}} = 0 \text{ km/h} \][/tex]
### Conclusion:
- The total distance covered during the round trip is [tex]\(480 \text{ km}\)[/tex].
- The total time taken for the round trip is [tex]\(48.5 \text{ hours}\)[/tex].
- The average speed of the cab is approximately [tex]\(9.897 \text{ km/h}\)[/tex].
- The average velocity of the cab is [tex]\(0 \text{ km/h}\)[/tex].
These results have been derived from the given data and provide a complete and precise solution to the problem.
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