Problem 35 - Algebra 1

Question

The models are based on data about train travel from 1990 to 1996 in the United States. Let \(t\) represent the number of years since \(1990 .\) D Source: Statistical Abstract of the United States Miles (in millions) traveled by passengers: \(\quad M=\frac{6300-800 t}{1-0.12 t}\) Passengers (in millions) who traveled by train: \(P=\frac{222-24 t}{10-t}\). Find a model for the average number of miles traveled per passenger.

Step-by-Step Solution

Answer

The model for the average number of miles traveled per passenger is \(A = \frac{6300-800t}{1-0.12t} * \frac{10-t}{222-24t}\)

Step 1: Understand the Problem

The problem provides two relations, \(M\) and \(P\), which represent miles traveled and passengers who traveled by train, respectively, as a function of time \(t\). It is required to find a model for the average miles traveled per passenger. This can be represented mathematically as \(A = \frac{M}{P}\), where \(A\) stands for the average miles travelled per passenger.

Step 2: Substitute the given functions into the formula

Substitute the expressions for \(M\) and \(P\) into the formula for average \(A\). It gives \(A = \frac{\frac{6300-800t}{1-0.12t}}{\frac{222-24t}{10-t}}\).

Step 3: Simplify the Expression

The formula to calculate ratio simplifies to \(A = \frac{6300-800t}{1-0.12t} * \frac{10-t}{222-24t}\). There are no common factors that can cancel each other out, so the expression can't be simplified any further.

Key Concepts

Functions and Rational Expressions

In mathematics, a function is a special relationship where each input has a single output. Rational functions are those that can be expressed as the ratio of two polynomial expressions. In our exercise, the functions provided are rational expressions dependent on the variable \( t \), which represents time in years since 1990.

Given:

• Miles traveled by passengers: \( M = \frac{6300 - 800t}{1 - 0.12t} \)
• Passengers: \( P = \frac{222 - 24t}{10 - t} \)

These functions describe how the miles traveled and the number of passengers change over time. By expressing these quantities as rational functions, we can easily model and predict behavior for various time periods.

Understanding Ratios

A ratio describes a relationship between two numbers, showing how many times the first number contains the second. In the context of this problem, we are dealing with the ratio of the total miles traveled to the total number of passengers, which gives us an average.

To find this, we use the formula for the average number of miles traveled per passenger, \( A = \frac{M}{P} \). This expression evaluates how each passenger contributes to the overall distance traveled. The key here is understanding that even though both \( M \) and \( P \) are complex expressions, the concept of dividing one by the other provides insight into the performance and trends in train travel over time.

• A ratio is not just a simple division; it's a comparison.
• This allows for better understanding of the context and meaning of both numerators and denominators.

Modeling with Mathematics

Mathematical modeling is a process where we use mathematical expressions to represent real-world situations. Here, rational functions are used to model the train travel data from 1990 to 1996. This approach allows us to evaluate and understand trends in an organized and calculable way.

When modeling with mathematics, especially through rational functions, it's essential to interpret the functions' behaviors:

• The numerator and denominator each tell part of the story in how they handle data;
• They can also guide predictions about unseen data within the parameter range.

By simplifying complex ratios as seen in this exercise, students can better grasp not only the mathematical basis but also the real-world implications of the data. This way of mathematical modeling is crucial for making informed decisions based on historical data and trends.