Road Network Travel Time Optimization

Question 1

Tyrion Lannister drove from Meereen to Oldtown, then from Oldtown to Lannisport, and finally from Lannisport to Winterfell, always taking the shortest paths. He always drove at a speed 10 km/hr below the maximum speed limits for the stretches he took. What was his total driving time closest to?

Accepted Answer

C. Distances: Meereen to Oldtown = 100 km. Reduced speed = $100 - 10 = 90$ km/hr. Time = $100/90 = 1.111$ hr. Oldtown to Lannisport = 60 km. Reduced speed = $20 - 10 = 10$ km/hr. Time = $60/10 = 6$ hr. Lannisport to Winterfell = 40 km. Reduced speed = $120 - 10 = 110$ km/hr. Time = $40/110 = 0.3636$ hr. Total time = $1.111 + 6 + 0.3636 = 7.475$ hrs = 7 hrs 28.5 mins.

Question 2

Missandei starts from Gulltown towards Oldtown by the shortest path, driving at the maximum permissible speed. From Oldtown, she drives at a speed of 10 km/hr towards Lannisport. When Missandei starts from Gulltown, Varys starts at the same time from Lannisport to Oldtown along the shortest path, always driving at the maximum permissible speed. If they don’t stop anywhere, at what point will they meet?

Accepted Answer

D. Missandei: Gulltown to Oldtown (50 km at 25 km/hr) takes 2 hours. In 2 hours, Varys travels from Lannisport towards Oldtown at 20 km/hr $\implies 20 	imes 2 = 40$ km. At $t = 2$ hours, Missandei is at Oldtown, Varys is 40 km north of Lannisport (20 km south of Oldtown). They now travel towards each other on the 20 km stretch between Oldtown and Varys's position: Relative speed = $10 + 20 = 30$ km/hr. Time to meet = $20 / 30 = 2/3$ hour. Distance Missandei travels from Oldtown = $10 	imes (2/3) = 6.67$ km. So they meet 6.67 km south of Oldtown.

Question 3

The capital city, King’s Landing, located 40 km to the south of Gulltown on the road connecting Gulltown to Winterfell, did not have a straight road connecting to Meereen. Now, a new expressway is being built to connect these two towns by a straight road. What should be the maximum speed limit allowed on this expressway so that it cuts down the travel time, from Meereen to King’s Landing, from the fastest possible route through the road network shown in the diagram, by 20 minutes?

Accepted Answer

D. Fastest route from Meereen to King's Landing currently: Path MWK: Meereen to Winterfell (120 km at 120 km/hr = 1 hr) + Winterfell to King's Landing (50 km at 120 km/hr = 25 mins) = 1 hr 25 mins = 85 mins. If expressway cuts time by 20 mins, new travel time = 65 mins = $13/12$ hours. Direct distance from Meereen to King's Landing = $\sqrt{120^2 + 50^2} = 130$ km. Required Speed = $130 / (13/12) = 120$ km/hr.

Road Network Travel Time Optimization

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