Question

Truck \(\mathrm{X}\) is 13 miles ahead of Truck \(\mathrm{Y}\), which is traveling the same direction along the same route as Truck X. If Truck \(\mathrm{X}\) is traveling at an average speed of 47 miles per hour and Truck \(Y\) is traveling at an average speed of 53 miles per hour, how long will it take Truck Y to overtake and drive 5 miles ahead of Truck X? A 2 hours B 2 hours 20 minutes C 2 hours 30 minutes D 2 hours 45 minutes E 3 hours

Short answer

3 hours

Step-by-step solution

- Define the variables

Let the time taken for Truck Y to overtake and drive 5 miles ahead of Truck X be \( t \) hours.

- Calculate the relative speed

Truck Y is traveling at 53 mph and Truck X is traveling at 47 mph. The relative speed of Truck Y with respect to Truck X is: \[ 53 \text{ mph} - 47 \text{ mph} = 6 \text{ mph} \]

- Calculate the distance to overtake

Truck Y needs to cover the 13 miles gap and an additional 5 miles to be ahead by 5 miles. So, the total distance Truck Y needs to cover is: \[ 13 \text{ miles} + 5 \text{ miles} = 18 \text{ miles} \]

- Use the formula for time

The time taken is given by distance divided by speed. Using the relative speed, the time taken for Truck Y to overtake and drive 5 miles ahead is: \[ t = \frac{18 \text{ miles}}{6 \text{ mph}} \]

- Solve for time

\[ t = 3 \text{ hours} \]

Key concepts

relative speed

When two objects move in the same direction, but at different speeds, the concept of relative speed helps us understand how quickly one object is changing their position with respect to the other. In this problem, Truck X and Truck Y are both traveling in the same direction, but Truck Y is faster.
The relative speed is simply the difference between their speeds:
Truck Y's speed - Truck X's speed = 53 mph - 47 mph = 6 mph.
This means that Truck Y is closing the gap between itself and Truck X at a rate of 6 miles per hour. Knowing the relative speed is crucial for calculating how long it will take one object to catch up and surpass another.

distance-time calculations

In problems like this one, we often need to determine how long it takes for one object to catch up to another. This can be solved by understanding the relationship of distance, speed, and time. The basic formula connecting these three is:
\[\text{time} = \frac{\text{distance}}{\text{speed}}\].
In our exercise, Truck Y needs to cover a total additional distance of 18 miles (13 miles to close the gap and 5 more miles to get ahead). We already know Truck Y's relative speed (6 mph). Hence, we use the formula:
\[\text{time} = \frac{18 \text{ miles}}{6 \text{ mph}} = 3 \text{ hours}\].
This tells us that it will take Truck Y 3 hours to not only catch up but also get 5 miles ahead of Truck X.

algebra

Algebra is fundamental in solving many distance-time problems. Setting up the right equations and variables can streamline solving such problems. Let's recap how it’s used here:
1. Define the variables: Let the time Truck Y needs to overtake and get 5 miles ahead of Truck X be t hours.
2. Use the data given: Truck Y speed: 53 mph, Truck X speed: 47 mph, and initial distance gap: 13 miles + 5 miles ahead.
3. Calculate the relative speed as 6 mph.
4. Use the formula for time based on the total distance and relative speed: \[\text{time} = \frac{18 \text{ miles}}{6 \text{ mph}}\].
By solving for t, which is 3 hours, we get the answer.
Understanding and setting up such equations is key in algebra to solve various real-world problems efficiently.