When Will the Train Be Full? A Mathematical Exploration

Introduction

Many of us have wondered at some point how many passengers a train can hold or when a train might be full given a certain pattern of passenger additions. This problem is a classic example of using mathematical concepts to solve real-world scenarios. In this article, we will explore how to determine the number of stops it takes for a train to be full under a specific pattern of passenger additions.

The Problem

The train has a capacity of 78 passengers and starts out empty. At each stop, the number of passengers increases by one. The first stop adds 1 passenger, the second stop adds 2 passengers, the third stop adds 3 passengers, and so on. How many stops will it take for the train to be full?

Making the Calculation

To find the number of stops required for the train to reach its full capacity, we need to understand the pattern and the mathematical formula underlying the problem. This can be approached as an arithmetic sequence problem combined with the use of a quadratic equation.

Arithmetic Sequence Approach

The number of passengers added at each stop forms an arithmetic sequence, where each term increases by 1 compared to the previous term. The sum of the first n terms of an arithmetic sequence can be calculated using the formula:

Sum (First Term Last Term) * Number of Terms / 2

In this case, the first term is 1 and the last term, which is also n, is the number of stops. The sum of the passengers needed to fill the train is 78.

Formulating the Equation

Putting it into the formula:

78 (1 n) * n / 2

Simplifying the equation:

156 n^2 - n

Bringing it to the standard form of a quadratic equation:

n^2 - n - 156 0

Solving the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

n (-b ± √(b2 - 4ac)) / 2a

In this equation, a 1, b -1, and c -156. Substituting these values in:

n (-(-1) ± √((-1)2 - 4 * 1 * (-156))) / (2 * 1)

Simplifying further:

n (1 ± √(1 624)) / 2

n (1 ± 25) / 2

Calculating the two potential values for n:

n (26 / 2) 13

n (-24 / 2) -12 (not applicable since n must be positive)

Thus, the number of stops required for the train to reach its full capacity is 12.

Conclusion

By applying the principles of arithmetic sequences and quadratic equations, we can accurately determine the number of stops it takes for a train to be full when passengers are added in an increasing pattern. The problem not only showcases the practical application of these mathematical concepts but also highlights their importance in real-world scenarios.

Understanding when a train is full is relevant for bus management and scheduling, ensuring efficient use of resources and optimizing service provision.