Problem 31 - Algebra 2

Question

The average NFL salary \(A(t)\) (in thousands of dollars) can be estimated using \(A(t)=2.3 t^{2}-12.4 t+73.7,\) where \(t\) is the number of years since 1975. Determine a domain and range for which this function makes sense.

Step-by-Step Solution

Answer

The domain is \([0, 30]\) and the range is \([73.7, 1517]\).

Step 1: Understand the Context of the Function

The function represents the average NFL salary in thousands of dollars with respect to years since 1975. This means that the input variable, \(t\), represents the number of years since 1975.

Step 2: Determine a Logical Domain

Since the function is a model of average NFL salaries starting from 1975, the domain is the set of all years after 1975 for which the data or prediction is relevant.Hence, the domain is \(t \geq 0\).

Step 3: Consider Practical Domain Limits

Since real-world models are usually relevant for a limited future, we assume the function is relevant for a number of years, say up to 30 years, making the arbitrary practical domain \(0 \leq t \leq 30\).

Step 4: Calculate the Range for Given Domain

The range can be determined by finding \(A(t)\) for boundary values of \(t\). Calculate \(A(0) = 2.3 \times 0^2 - 12.4 \times 0 + 73.7 = 73.7\).Calculate \(A(30) = 2.3 \times 30^2 - 12.4 \times 30 + 73.7 = 1517\). Thus, the range is \([73.7, 1517]\).

Step 5: Conclude the Domain and Range

Based on calculations and context, the domain (in years since 1975) can be taken as \([0, 30]\), and the range of the average salary \(A(t)\) is \([73.7, 1517]\).

Key Concepts

Domain and Range

When dealing with quadratic functions such as our NFL salary model, understanding the domain and range is essential to interpreting data accurately. For the given function, the domain refers to all possible values that the variable \( t \) can take. Here, \( t \) represents the number of years since 1975. Since salaries can't be measured for a time before the start point, the earliest \( t \) is 0, corresponding to the year 1975. Calculating the likely relevance of this model, experts often impose a limit, like 30 years, to avoid unrealistic projections. Thus, the practical domain becomes \( 0 \leq t \leq 30 \).
The range, on the other hand, outlines all possible output values of the function. It is determined by calculating the salary at boundary years of our domain. At \( t = 0 \), the salary was \( 73.7 \) thousand dollars, and at \( t = 30 \), it jumps to \( 1517 \) thousand dollars. As such, the estimated range of this model is \([73.7, 1517]\). This range represents the span of average NFL salaries during the considered timeframe. In essence, domain and range provide the scope of input years and output salaries, guiding analysts in interpreting what the function realistically represents.

Mathematical Modelling

Mathematical modelling is a powerful tool that helps translate real-world situations into mathematical equations or functions. In this instance, the average NFL salary model is depicted using a quadratic equation, \( A(t) = 2.3t^2 - 12.4t + 73.7 \). This type of equation is known for having a parabolic graph shape, characterized by a curve that can represent various trends or phenomena in everyday life.
The choice of a quadratic function in this context might arise from certain trends observed in salaries over time, such as initial rises and eventual stabilization or further increase after achieving a peak. These trends can often result in a parabolic curve, with the function parameters adjusted to best fit the historical salary data.
The use of such a model allows stakeholders to forecast future salaries, making it a potent tool in strategic planning and financial management. By inputting different values of \( t \), one can estimate the corresponding salary at a specific year post-1975. Thus, mathematical models serve both as predictors and as interpreters of historical data, aiding in decision-making processes.

Real-World Applications

Quadratic functions have numerous real-world applications beyond our NFL salary model example. These functions are essential in various fields due to their ability to represent complex relationships succinctly.

• **Economics:** Quadratic equations can simulate cost, revenue, and profit maximization scenarios. Here, they help businesses understand how changing variables affect financial outcomes.
• **Physics and Engineering:** Parabolic motion, projectile trajectories, and optimization problems frequently use quadratic functions to provide accurate predictions and solutions.
• **Biology and Ecology:** Quadratics model population growth and genetic variations, offering insights into species development over time.

In the context of our NFL salary application, quadratic models offer a method to gauge and predict financial trends in sports, affecting decisions about player contracts, salary caps, and team budgeting. Analysts utilize these models to estimate future expenses, ensuring that organizations can plan accordingly in a dynamic economic environment. Hence, quadratic functions encapsulate real-world dynamics, bringing clarity and foresight to scenarios across various domains.