Problem 83 - College Algebra

Question

The annual amount that we spend to attend sporting events can be modeled by $$f(x)=2.05+1.3 \ln x$$ where \(x\) represents the number of years after 1984 and \(f(x)\) represents the total annual expenditures for admission to spectator sports, in billions of dollars. In \(2000,\) approximately how much was spent on admission to spectator sports?

Step-by-Step Solution

Answer

In the year 2000, approximately $6.53 billion were spent on admission to spectator sports.

Step 1: Find the Value for \(x\)

First, have to find the value of \(x\). In this context, \(x\) refers to the number of years after 1984. Since the year in question is 2000, we can calculate \(x\) by subtracting 1984 from 2000, which gives us \(x = 16\).

Step 2: Substituting \(x\) into the function

Next, we need to substitute this \(x\) value into the given function, \(f(x)=2.05+1.3 \ln x\), to calculate the total annual expenditures on admission. Substituting \(x = 16\) into the function gives us \(f(16) = 2.05 + 1.3 \ln 16\).

Step 3: Solve the equation

Finally, we need to solve the function for the given \(x\) value. Using the logarithmic property we simplify the equation and get the final answer: \(f(16) = 2.05 + 1.3 \ln 16 \approx 6.53\)

Key Concepts

College Algebra

College Algebra is a foundational course that shapes a student’s mathematical prowess and problem-solving skills. Its applications range from simple equations to complex functions, including the logarithmic functions seen in the given exercise. To understand a function like f(x) = 2.05 + 1.3 \( ln x \), it's essential to grasp how algebraic expressions and functions operate.

Students learn to work with variables, which represent numbers, and how these can change within an equation or function. Calculating the value of a function given a specific input—like determining the spending on sports events in a certain year based on the given model—is a prime example of applying algebraic principles. The ability to manipulate and solve equations is key to finding solutions in algebra, as exemplified by substituting the number of years into the logarithmic function to predict expenditures.

Algebraic Modeling

Algebraic modeling is a powerful technique used in College Algebra to represent real-world situations with mathematical expressions or functions. The exercise provided utilizes a model f(x)=2.05 + 1.3 \( ln x \) to relate the passage of years to the expenditures for sporting events. By defining x as the number of years after 1984, the equation gains context and meaning, transforming it into a tool for prediction and analysis.

Understanding how to construct these models and interpret their results is crucial. For example, the use of the natural logarithm within our model captures a specific kind of growth pattern in spending, and adjusting coefficients like 2.05 and 1.3 allows for refining the model’s accuracy. This practical application of mathematics is invaluable across various fields, such as economics, engineering, and the sciences.

Natural Logarithms

Natural logarithms are a cornerstone concept in College Algebra, particularly when it comes to understanding growth processes and time-based changes. The natural logarithm, symbolized as ln, is the logarithm to the base \( e \), where e is an irrational and transcendental number approximately equal to 2.71828.

When dealing with compound interest, population growth, or, as in our exercise, the growth of spending over time, natural logarithms are incredibly useful. The logarithmic function in the model, \( 1.3 \, ln x \), implies that the increase in spending does not follow a simple linear pattern but rather a logarithmic one, which typically represents a slowing growth rate over time. This insight is essential when interpreting the model's outcomes and using them to make predictions. Therefore, mastering the use of natural logarithms is essential for students in algebraic modeling.