Answer :
Let's solve the problem step by step.
1. Total time of the journey:
[tex]\[ \text{Total time} = 5 \text{ hours} \][/tex]
2. Fraction of the journey in the first segment:
[tex]\[ \text{First segment fraction} = \frac{2}{3} \][/tex]
3. Time spent on the first segment:
[tex]\[ \text{Time for first segment} = \frac{2}{3} \times 5 \text{ hours} \][/tex]
[tex]\[ \text{Time for first segment} = \frac{10}{3} \text{ hours} \approx 3.3333 \text{ hours} \][/tex]
4. Time spent on the second segment:
[tex]\[ \text{Time for second segment} = 5 - \frac{10}{3} \text{ hours} \][/tex]
[tex]\[ \text{Time for second segment} = \frac{15}{3} - \frac{10}{3} \text{ hours} = \frac{5}{3} \text{ hours} \approx 1.6667 \text{ hours} \][/tex]
5. Distance covered in the first segment:
[tex]\[ \text{Speed for first segment} = 40 \text{ km/hr} \][/tex]
[tex]\[ \text{Distance for first segment} = 40 \times \frac{10}{3} \text{ hours} \][/tex]
[tex]\[ \text{Distance for first segment} = \frac{400}{3} \text{ km} \approx 133.3333 \text{ km} \][/tex]
6. Distance covered in the second segment:
[tex]\[ \text{Speed for second segment} = 60 \text{ km/hr} \][/tex]
[tex]\[ \text{Distance for second segment} = 60 \times \frac{5}{3} \text{ hours} \][/tex]
[tex]\[ \text{Distance for second segment} = \frac{300}{3} \text{ km} = 100 \text{ km} \][/tex]
7. Total distance of the journey:
[tex]\[ \text{Total distance} = \text{Distance for first segment} + \text{Distance for second segment} \][/tex]
[tex]\[ \text{Total distance} = 133.3333 \text{ km} + 100 \text{ km} \][/tex]
[tex]\[ \text{Total distance} = 233.3333 \text{ km} \][/tex]
Thus, the length of the journey is approximately [tex]\( 233.3333 \)[/tex] km, which corresponds to the distance calculation we've performed.
Therefore, none of the options provided in the problem statement exactly match the calculated value of [tex]\( 233.3333 \)[/tex] km. The closest match in this scenario, given no exact options, should be:
[tex]\[ \boxed{233 \text{ km}} \][/tex]
1. Total time of the journey:
[tex]\[ \text{Total time} = 5 \text{ hours} \][/tex]
2. Fraction of the journey in the first segment:
[tex]\[ \text{First segment fraction} = \frac{2}{3} \][/tex]
3. Time spent on the first segment:
[tex]\[ \text{Time for first segment} = \frac{2}{3} \times 5 \text{ hours} \][/tex]
[tex]\[ \text{Time for first segment} = \frac{10}{3} \text{ hours} \approx 3.3333 \text{ hours} \][/tex]
4. Time spent on the second segment:
[tex]\[ \text{Time for second segment} = 5 - \frac{10}{3} \text{ hours} \][/tex]
[tex]\[ \text{Time for second segment} = \frac{15}{3} - \frac{10}{3} \text{ hours} = \frac{5}{3} \text{ hours} \approx 1.6667 \text{ hours} \][/tex]
5. Distance covered in the first segment:
[tex]\[ \text{Speed for first segment} = 40 \text{ km/hr} \][/tex]
[tex]\[ \text{Distance for first segment} = 40 \times \frac{10}{3} \text{ hours} \][/tex]
[tex]\[ \text{Distance for first segment} = \frac{400}{3} \text{ km} \approx 133.3333 \text{ km} \][/tex]
6. Distance covered in the second segment:
[tex]\[ \text{Speed for second segment} = 60 \text{ km/hr} \][/tex]
[tex]\[ \text{Distance for second segment} = 60 \times \frac{5}{3} \text{ hours} \][/tex]
[tex]\[ \text{Distance for second segment} = \frac{300}{3} \text{ km} = 100 \text{ km} \][/tex]
7. Total distance of the journey:
[tex]\[ \text{Total distance} = \text{Distance for first segment} + \text{Distance for second segment} \][/tex]
[tex]\[ \text{Total distance} = 133.3333 \text{ km} + 100 \text{ km} \][/tex]
[tex]\[ \text{Total distance} = 233.3333 \text{ km} \][/tex]
Thus, the length of the journey is approximately [tex]\( 233.3333 \)[/tex] km, which corresponds to the distance calculation we've performed.
Therefore, none of the options provided in the problem statement exactly match the calculated value of [tex]\( 233.3333 \)[/tex] km. The closest match in this scenario, given no exact options, should be:
[tex]\[ \boxed{233 \text{ km}} \][/tex]
