Answer:
Time taken by Duke on the highway = 3 hours
Time spent on the construction = 6 hours
Time spent in the city = 2 hours
Step-by-step explanation:
Speed by which Duke travels a distance on highway
= 60 mph
Let the time taken to travel on highway =
hours
Distance traveled =
miles
Speed in the city = 40 mph
Let the time taken =
hours
Distance traveled in the city =
miles
Speed on the construction = 20 mph
Let the time spend on the construction =
hours
Distance traveled on the construction =
miles
Since, total distance covered = 380 miles
![d_1+d_2+d_3=60t_1+40t_2+20t_3](https://tex.z-dn.net/?f=d_1%2Bd_2%2Bd_3%3D60t_1%2B40t_2%2B20t_3)
He spends one half the time on the highway as he does on the construction,
![t_3=2t_1](https://tex.z-dn.net/?f=t_3%3D2t_1)
Total time spent in the trip = 11 hours
Therefore, ![t_1+t_2+t_3=11](https://tex.z-dn.net/?f=t_1%2Bt_2%2Bt_3%3D11)
![t_1+t_2+2t_1=11](https://tex.z-dn.net/?f=t_1%2Bt_2%2B2t_1%3D11)
![t_2=11-3t_1](https://tex.z-dn.net/?f=t_2%3D11-3t_1)
From equation (1),
![60t_1+40(11-3t_1)+20(2t_1)=380](https://tex.z-dn.net/?f=60t_1%2B40%2811-3t_1%29%2B20%282t_1%29%3D380)
![60t_1-120t_1+40t_1+440=380](https://tex.z-dn.net/?f=60t_1-120t_1%2B40t_1%2B440%3D380)
![440-20t_1=380](https://tex.z-dn.net/?f=440-20t_1%3D380)
![20t_1=440-380](https://tex.z-dn.net/?f=20t_1%3D440-380)
hours
hours
hours
Therefore, time taken by Duke on the highway = 3 hours
Time spent on the construction = 6 hours
Time spent in the city = 2 hours