- Express the distance traveled by each car using the formula \(\text{distance} = \text{rate} \times \text{time}\).
- Note that the first car travels \(60t\) miles and the second travels \(55t\) miles.
- Since the cars move toward each other and meet at a total distance of \(150\) miles, equate the sum of their individual distances to the total initial distance.
- The resulting equation is \(\boxed{60t + 55t = 150}\).
Explanation
- Analyzing the Given Information
Two cars start at an initial distance of \(150\) miles from each other and travel toward one another. Let \(t\) represent the time in hours it takes for the two cars to pass each other. We are given the following information:
- The speed of the first car is \(60\) miles per hour.
- The speed of the second car is \(55\) miles per hour.
- The time each car travels before meeting is \(t\).
Since distance is the product of rate and time (\(d = r \times t\)), the distance traveled by the first car is \(60 \times t = 60t\) miles, and the distance traveled by the second car is \(55 \times t = 55t\) miles.
- Formulating the Equation
When the two cars pass each other, they will have covered a total combined distance equal to their initial starting distance. This is because they start at two different locations and move toward each other until their paths intersect. Thus, the sum of the distances traveled by both cars must equal the initial gap between them:
\[\text{Total distance} = \text{Distance of first car} + \text{Distance of second car}\]
Substituting the expressions derived in the previous step:
\[150 = 60t + 55t\]
Alternatively written as:
\[60t + 55t = 150\]
This equation allows us to solve for the time \(t\).
- Concluding the Answer
Comparing the derived equation \(60t + 55t = 150\) with the given options:
- \(60t - 55t = 0\)
- \(60t + 55t = 1\)
- \(60t + 55t = 150\)
- \(60t - 55t = 150\)
The third option matches our derived equation perfectly. Therefore, the equation to determine the time \(t\) is \(60t + 55t = 150\). The final answer is \(\boxed{60t + 55t = 150}\).
Examples
This problem is a classic example of relative motion. Similar principles are used by air traffic controllers or logistics managers to calculate arrival times, ensure safety, and coordinate movements of multiple objects or vehicles. For example, if two trains are on parallel tracks headed toward each other, knowing their speeds and the track distance allows you to predict exactly when they will cross paths, which is essential for scheduling and safety.