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Difficulty: MediumAlgebraic Word Problems and Modeling

A logistics company operates a delivery van and a cargo drone along a straight route of 270270 kilometers connecting Hub X and Hub Y. The delivery van departs from Hub X toward Hub Y at a constant speed of 6060 kilometers per hour. Exactly 22 hours later, the cargo drone departs from Hub Y toward Hub X along the same route at a constant speed of 9090 kilometers per hour. How many hours after the delivery van departs will the van and the cargo drone meet?

  1. A
    1.8 hours
  2. B
    2.6 hours
  3. 3.0 hoursAnswer
  4. D
    4.5 hours
  5. E
    5.0 hours

Answer

3.0 hours
The option specifying '3.0 hours' is correct. During the 2 hours before the drone departs, the van covers 60 km/h×2 h=120 km60 \text{ km/h} \times 2 \text{ h} = 120 \text{ km}. The remaining distance between the vehicles is 270 km−120 km=150 km270 \text{ km} - 120 \text{ km} = 150 \text{ km}. Once the drone departs, the two vehicles close the gap at a combined rate of 60+90=150 km/h60 + 90 = 150 \text{ km/h}. The time needed to cover the remaining 150 km150 \text{ km} is 150 km150 km/h=1 hour\frac{150 \text{ km}}{150 \text{ km/h}} = 1 \text{ hour}. Adding the van's initial 22 hours gives a total travel time of 3.03.0 hours.

Step-by-Step Solution

1
Define variables for travel time and compute distance covered by the van before the drone departs.
In 22 hours at 6060 km/h, the van covers 60×2=12060 \times 2 = 120 kilometers.
The van travels alone for the first two hours.
2
Determine the remaining distance separating the two vehicles when the drone starts moving.
Remaining distance = 270−120=150270 - 120 = 150 kilometers.
The total distance between the hubs is 270270 kilometers.
3
Calculate the relative speed of approach and the additional time required to meet.
Combined speed = 60+90=15060 + 90 = 150 km/h. Additional time = 150150=1\frac{150}{150} = 1 hour.
Since the vehicles travel toward each other, their speeds add up.
4
Find the total time elapsed since the van departed.
Total time = 2+1=3.02 + 1 = 3.0 hours.
The question asks for total hours since the van's departure.

Key Concept

Distance-Rate-Time linear modeling with staggered start times and opposing directions
Estimated Time:1m 45s
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Topics

Algebraic Word Problems and Modeling

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