Understanding the Best Use of Variable Speeds in Travel

In planning a journey, optimal use of different speeds can significantly impact the total travel time, especially when considering the distance and time constraints. This article discusses a practical example where a car travels a 130-mile trip at varying speeds—is the journey optimized for time efficiency or is there a more effective strategy?

The Problem and Solution Approach

A common scenario might be encountered where a car travels part of the distance at one speed and the remainder at a different speed. In this example, a 130-mile trip was completed in 2.5 hours, traveling at 55 mph for part of the time and reducing the speed to 40 mph for the remainder. The question is: for how long did the car travel at 40 mph?

System of Equations to Solve the Problem

To solve this problem, we can use a system of equations based on the information and time available. Let's define:

t1 as the time traveled at 55 mph t2 as the time traveled at 40 mph

According to the problem, the total time for the trip is 2.5 hours:

Equation 1:

t1 t2 2.5

Next, we can express the distances traveled at each speed:

Distance at 55 mph: d1 55t1 Distance at 40 mph: d2 40t2 Total distance: d1 d2 130

By substituting these expressions into the total distance equation, we get:

Equation 2:

55t1 40t2 130

Now, substituting the first equation into the second to solve for t2:

t1 2.5 - t2 Substitute into the distance equation:

55(2.5 - t2) 40t2 130

Expanding and simplifying:

Equation 3:

137.5 - 55t2 40t2 130

Combining like terms:

137.5 - 15t2 130

Solving for t2:

-15t2 -7.5

t2 0.5 hours or 30 minutes

Therefore, the car traveled at 40 mph for 0.5 hours or 30 minutes.

Additional Insights: Optimal Travel Time

By understanding the travel times at each speed and the distances covered, we can further optimize future trips. For those who frequently travel long distances, calculating the best speed to use at different stages of the journey can lead to time savings and fuel efficiency.

Conclusion and Further Exploration

This mathematical breakdown demonstrates the practical application of algebra in real-life problems. It shows how setting up and solving a system of equations can help answer questions related to travel time, distance, and speed. Such problems can be adapted to various scenarios, from short commutes to long-distance driving.

For future reference, one might consider how varying speeds affects not only time but also fuel consumption. This exploration opens the door to considering more complex models, like using traffic data or alternative routes to minimize travel time even further.

Related Keywords:

optimal travel time variable speeds math problem solving