A scout rover leaves a forward operating base and travels along a straight path at a const… — Armed Services Vocational Aptitude Battery (ASVAB)

A scout rover leaves a forward operating base and travels along a straight path at a constant speed of $40$ mph. Exactly $30$ minutes later, a heavily armored command vehicle leaves the same base along the same path at a constant speed of $55$ mph to overtake the rover. How many minutes will the command vehicle need to drive before it catches up to the scout rover?

Answer: 80 / 80 minutes / 80 mins / eighty

Answer

The scout rover establishes a 20-mile lead during its 30-minute (0.5 hour) head start. The command vehicle closes this 20-mile gap at a relative speed of 15 mph (55 mph - 40 mph). It takes 4/3 hours to close the gap, which converts to exactly 80 minutes.

Step-by-Step Solution

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

30

minutes is equal to

0.5

hours.

Speed is given in miles per hour (mph), so time must be in hours to accurately calculate the distance.

Calculate the head start distance of the scout rover.

40 \text{ mph} \times 0.5 \text{ hours} = 20 \text{ miles}

This establishes the initial gap that the command vehicle must close.

Calculate the relative catch-up speed.

55 \text{ mph} - 40 \text{ mph} = 15 \text{ mph}

Since both vehicles are traveling in the same direction, subtract the slower speed from the faster speed to find how fast the gap is closing.

Calculate the time required to catch up in hours.

20 \text{ miles} \div 15 \text{ mph} = \frac{4}{3} \text{ hours}

Time equals distance divided by rate.

Convert the catch-up time back to minutes.

\frac{4}{3} \times 60 = 80 \text{ minutes}

The question specifically asks for the driving time in minutes.

Key Concept

Relative Motion and Catch-Up Scenarios

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