A logistics company dispatches a delivery van traveling on a straight highway at a constan… — Armed Services Vocational Aptitude Battery (ASVAB)

A logistics company dispatches a delivery van traveling on a straight highway at a constant speed of $40$ mph. Exactly $90$ minutes later, a courier is dispatched from the same location to intercept the van, traveling on the same highway at a constant speed of $60$ mph. How many hours will it take the courier to catch up to the delivery van after departing?

A
$4.5$
B
$0.6$
$3$ Answer
D
$1.8$

Answer

It will take

3

hours for the courier to catch up to the delivery van.

First, the van's head start of 90 minutes must be converted to 1.5 hours. Traveling at 40 mph for 1.5 hours puts the van 60 miles ahead of the starting point. Next, because the vehicles travel in the same direction, their relative speed is the difference between their individual speeds: 60 mph - 40 mph = 20 mph. This means the courier gains 20 miles on the van every hour. Finally, divide the 60-mile gap by the 20 mph relative speed to find the time: 60 / 20 = 3 hours.

Step-by-Step Solution

Convert the van's head start time from minutes to hours.

90 \text{ minutes} = 1.5 \text{ hours}

Speed is given in miles per hour (mph), so all time units must be in hours to maintain consistency.

Calculate the initial distance gap created by the van's head start.

40 \text{ mph} \times 1.5 \text{ hours} = 60 \text{ miles}

Distance equals rate multiplied by time. This is the gap the courier must close.

Determine the relative speed of the courier closing the gap.

60 \text{ mph} - 40 \text{ mph} = 20 \text{ mph}

Because both vehicles are moving in the same direction, subtract the van's speed from the courier's speed to find how fast the distance between them is shrinking.

Calculate the time required for the courier to close the initial gap.

60 \text{ miles} / 20 \text{ mph} = 3 \text{ hours}

Time equals distance divided by rate. Dividing the initial gap by the relative speed gives the catch-up time.

Key Concept

Relative motion in the same direction requires subtracting the speed of the slower object from the faster object to find the relative rate of closure.

Estimated Time:1m 30s

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