Answer :
To find the average speed for the car's total journey, we need to take into account the total distance traveled and the total time spent traveling.
### Step-by-Step Solution:
1. Define Variables and Assumptions:
- Let [tex]\( d \)[/tex] be the distance (in miles) that the car travels in one direction.
- The speed while traveling to the destination: 40 mph.
- The speed while returning: 60 mph.
2. Calculate Time Taken for Each Segment:
- Time taken to travel to the destination:
[tex]\[ \text{Time}_{going} = \frac{d}{40} \text{ hours} \][/tex]
- Time taken to return:
[tex]\[ \text{Time}_{returning} = \frac{d}{60} \text{ hours} \][/tex]
3. Calculate Total Distance Traveled:
- The total distance is the sum of the distance going and the distance returning:
[tex]\[ \text{Total Distance} = d + d = 2d \text{ miles} \][/tex]
4. Calculate Total Time Taken:
- The total time is the sum of the time taken to go to the destination and the time taken to return:
[tex]\[ \text{Total Time} = \frac{d}{40} + \frac{d}{60} \][/tex]
5. Simplify the Total Time Taken:
- To simplify the expression for total time, find a common denominator. The least common multiple of 40 and 60 is 120.
[tex]\[ \frac{d}{40} = \frac{3d}{120} \][/tex]
[tex]\[ \frac{d}{60} = \frac{2d}{120} \][/tex]
- Adding these fractions:
[tex]\[ \text{Total Time} = \frac{3d}{120} + \frac{2d}{120} = \frac{5d}{120} = \frac{d}{24} \text{ hours} \][/tex]
6. Calculate Average Speed:
- Average speed is defined as the total distance divided by the total time:
[tex]\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{24}} = 2d \times \frac{24}{d} = 2 \times 24 = 48 \text{ mph} \][/tex]
The average speed for the total journey is 48 mph.
### Step-by-Step Solution:
1. Define Variables and Assumptions:
- Let [tex]\( d \)[/tex] be the distance (in miles) that the car travels in one direction.
- The speed while traveling to the destination: 40 mph.
- The speed while returning: 60 mph.
2. Calculate Time Taken for Each Segment:
- Time taken to travel to the destination:
[tex]\[ \text{Time}_{going} = \frac{d}{40} \text{ hours} \][/tex]
- Time taken to return:
[tex]\[ \text{Time}_{returning} = \frac{d}{60} \text{ hours} \][/tex]
3. Calculate Total Distance Traveled:
- The total distance is the sum of the distance going and the distance returning:
[tex]\[ \text{Total Distance} = d + d = 2d \text{ miles} \][/tex]
4. Calculate Total Time Taken:
- The total time is the sum of the time taken to go to the destination and the time taken to return:
[tex]\[ \text{Total Time} = \frac{d}{40} + \frac{d}{60} \][/tex]
5. Simplify the Total Time Taken:
- To simplify the expression for total time, find a common denominator. The least common multiple of 40 and 60 is 120.
[tex]\[ \frac{d}{40} = \frac{3d}{120} \][/tex]
[tex]\[ \frac{d}{60} = \frac{2d}{120} \][/tex]
- Adding these fractions:
[tex]\[ \text{Total Time} = \frac{3d}{120} + \frac{2d}{120} = \frac{5d}{120} = \frac{d}{24} \text{ hours} \][/tex]
6. Calculate Average Speed:
- Average speed is defined as the total distance divided by the total time:
[tex]\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{24}} = 2d \times \frac{24}{d} = 2 \times 24 = 48 \text{ mph} \][/tex]
The average speed for the total journey is 48 mph.
