Problem 59 - Algebra 1: Concepts and Skills

Question

Use the example on the previous page as a model. The average salary \(s\) (in thousands) for a professional baseball player in the United States can be modeled by \(s=136(1.18)^{t}\) where \(t=0\) represents the year \(1980 .\) Find the ratio of the average salary in 1985 to the average salary in \(1990 .\)

Step-by-Step Solution

Answer

The ratio of the average salary in 1985 to the average salary in 1990 is the result of the calculation in Step 3.

Step 1: Find the average salary in 1985

As \(t\) is the no of years after 1980, the value of \(t\) for the year 1985 is \(1985-1980=5\). Substitute \(t=5\) into the average salary equation \(s=136(1.18)^{t}\) to get the average salary in 1985.

Step 2: Find the average salary in 1990

The value of \(t\) for the year 1990 is \(1990-1980=10\). Substitute \(t=10\) into the salary equation \(s=136(1.18)^{t}\) to get the average salary in 1990.

Step 3: Compute the ratio of salaries

With the average salaries from 1985 (Step 1) and 1990 (Step 2), divide the salary of 1985 by the salary of 1990 to calculate their ratio.

Key Concepts

Average Salary Calculation

Understanding how to calculate average salary is important for a variety of financial assessments. For example, when examining the salary of a professional baseball player over time, we can use a specific equation that takes into account the base salary and the growth rate over a given period. In practice, to calculate the average salary at a given year, you substitute the time variable with the number of years that have pased since a base year.

To illustrate, let's look at our exercise: the average salary equation of a baseball player is given as \( s = 136(1.18)^t \), where \( s \) is in thousands of dollars and \( t \) is the number of years after 1980. If you're asked to compute the salary in 1985, you would find \( t \) by subtracting 1980 from 1985, getting 5 years, and substitute it into the equation to get the average salary for that year.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are essential for modeling growth processes or decay that occur in natural, social, and economic phenomena. The exponential growth model, in particular, fits scenarios where growth rate is proportional to the current value, leading to the rapid increase of the quantity over time.

In the given formula \( s = 136(1.18)^t \), the base is 1.18, which represents a consistent annual growth rate of 18%. As time \( t \) increases, the salary grows exponentially. This characteristic rapid growth is why exponential functions are used to describe scenarios from population dynamics to compounding interest in finance.

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world situations. These models help us analyze and make predictions based on the behavior of the modeled system. When creating a mathematical model, such as the one for calculating average salaries over time, there are several steps to follow:

• Identify the variables and constants that affect the system.
• Establish the relationships between those variables and constants, often using historical data.
• Translate these relationships into a mathematical formula that can predict future outcomes.

Effective modeling requires both mathematical understanding and deep knowledge of the system being modeled. In the case of a baseball player’s salary, a model accounted for historical salary trends to predict future changes. As models are simplified representations, their predictions are subject to the accuracy of assumptions and historical data.