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Difficulty: HardTranslating and Solving Algebraic Word Problems

Two courier drones, Drone A and Drone B, fly in opposite directions along a straight path from the same distribution center. Drone A departs at 10:00 AM and flies at a constant speed of 40 miles per hour40\text{ miles per hour}. Drone B departs from the same location at 10:30 AM and flies in the opposite direction at a constant speed of 60 miles per hour60\text{ miles per hour}. At what time will the two drones be exactly 220 miles220\text{ miles} apart?

  1. A
    12:00 PM
  2. B
    12:12 PM
  3. 12:30 PMAnswer
  4. D
    12:42 PM
  5. E
    12:54 PM

Answer

12:30 PM
The correct answer is 12:30 PM. To find this, define tt as the travel time of Drone B in hours. Since Drone A departs 30 minutes (0.50.5 hours) earlier, its travel time is t+0.5t + 0.5 hours. Using the relationship that the sum of the distances traveled in opposite directions equals the total distance, we set up the equation 40(t+0.5)+60t=22040(t + 0.5) + 60t = 220. Solving this yields 100t+20=220100t + 20 = 220, which simplifies to 100t=200100t = 200 and t=2 hourst = 2\text{ hours}. Adding 2 hours to Drone B's departure time of 10:30 AM results in 12:30 PM.

Step-by-Step Solution

1
Define variables for the travel times of both drones relative to their departure times.
Let tt be the number of hours Drone B travels after departing at 10:30 AM. Since Drone A departs 30 minutes (0.50.5 hours) earlier at 10:00 AM, Drone A's travel time is t+0.5t + 0.5 hours.
Establishing a single variable for time allows us to write a single-variable linear equation for the total distance covered.
2
Express the distance traveled by each drone using the formula Distance=Speed×Time\text{Distance} = \text{Speed} \times \text{Time} and sum them to equal the total separation distance.
The distance traveled by Drone A is 40(t+0.5)40(t + 0.5) miles, and the distance traveled by Drone B is 60t60t miles. The equation is: 40(t+0.5)+60t=22040(t + 0.5) + 60t = 220.
Since the drones fly in opposite directions from the same starting point, the total distance between them is the sum of their individual distances.
3
Solve the equation for tt.
40t+20+60t=220  ⟹  100t+20=220  ⟹  100t=200  ⟹  t=2 hours40t + 20 + 60t = 220 \implies 100t + 20 = 220 \implies 100t = 200 \implies t = 2\text{ hours}.
Solving for tt gives the number of hours Drone B travels before they are 220 miles220\text{ miles} apart.
4
Determine the final time by adding the calculated travel time to the corresponding departure time.
Drone B departs at 10:30 AM and travels for 2 hours2\text{ hours}. 10:30 AM+2 hours=12:30 PM10:30\text{ AM} + 2\text{ hours} = 12:30\text{ PM}. (Alternatively, Drone A departs at 10:00 AM and travels for 2+0.5=2.5 hours2 + 0.5 = 2.5\text{ hours}, which also yields 12:30 PM).
Adding the elapsed time to the initial departure time gives the clock time when the condition is met.

Key Concept

Translating relative motion and time-offset word problems into linear equations
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Topics

Translating and Solving Algebraic Word Problems

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