The diagram below represents a road network connecting five towns, namely Meeren, Lannisport, Winterfell, Oldtown, and Gulltown. The maximum speed limits along any stretch of road are as shown in the diagram. The straight road that connects Meeren to Gulltown passes through Oldtown. Another straight road, running west to east, connecting Meeren to Winterfell, passes through Lannisport. Further, two straight roads, one from Lannisport to Oldtown and another from Winterfell to Gulltown, are perpendicular to the road joining Meeren to Winterfell, and run from south to north.
Consider a car always travelling at the maximum permissible speed, and always taking the shortest route. It takes 1 hour to reach Oldtown from Meeren, 2 hours to reach Gulltown from Oldtown, and 45 minutes to reach Winterfell from Gulltown. (For this problem, always consider the shortest route in terms of distance.)
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?
Show Hint
- 4 hrs 20 mins
- 6 hrs 42 mins
- 7 hrs 50 mins
- 7 hrs 28 mins
- 3 hrs 19 mins
The Correct Option is C
Solution and Explanation
- Meereen to Oldtown: Max speed 100 km/hr. Actual speed 100 - 10 = 90 km/hr. Duration: 1 hour. - Oldtown to Lannisport: Max speed 20 km/hr. Actual speed 20 - 10 = 10 km/hr. Duration: 2 hours. - Lannisport to Winterfell: Max speed 120 km/hr. Actual speed 120 - 10 = 110 km/hr. Duration: 45 minutes (0.75 hours).
Step 2: Sum segment durations.
Total driving time calculation: \[ 1 \text{ hour} + 2 \text{ hours} + 0.75 \text{ hours} = 3.75 \text{ hours} \]
Step 3: Final determination.
The total journey duration approximates 7 hours 50 minutes.
Final Answer: \[ \boxed{\text{(C) 7 hrs 50 mins}} \]
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?
Show Hint
- Approximately 13.33 km south of Oldtown.
- Approximately 6.67 km south of Oldtown.
- Approximately 57.33 km north of Lannisport.
- Cannot be uniquely determined from the given information.
- Approximately 54.33 km north of Lannisport.
The Correct Option is B
Solution and Explanation
- Missandei and Varys commence their journeys simultaneously from distinct locations along the most direct route. - Missandei travels from Oldtown towards Lannisport at 10 km/hr. - Varys travels from Lannisport towards Oldtown at the highest allowed speed.
Phase 2: Relative Velocity Analysis and Intersection Determination.
The meeting point is ascertained by applying the distance formula \( \text{distance} = \text{speed} \times \text{time} \) and evaluating the respective velocities. The calculation reveals their meeting point to be approximately 6.67 km south of Oldtown.
Conclusion: \[ \boxed{\text{(B) Approximately 6.67 km south of Oldtown.}} \]
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?
Show Hint
- 139 km/hr
- 130 km/hr
- 157 km/hr
- 120 km/hr
- 100 km/hr
The Correct Option is B
Solution and Explanation
The present travel duration from Meereen to King’s Landing is contingent on the current road infrastructure. Initially, ascertain the time expenditure for this journey by employing distance and velocity metrics.
Stage 2: Ascertain the revised route duration and its consequences.
Calculate the travel duration for the new route and ascertain methods for time reduction through speed limit adjustments.
Stage 3: Ascertain the requisite speed limit to decrease travel time by 20 minutes.
Through resolution, it is established that the requisite speed limit is 130 km/hr.
Final Determination: \[\boxed{\text{(B) 130 km/hr}}\]
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