Problem 74 - Algebra 2

Question

In \(1990,2,573,225\) people attended St. Louis Cardinals home games. In 2004 , the attendance was \(3,048,427 .\) What was the average annual rate of increase in attendance?

Step-by-Step Solution

Answer

The average annual rate of increase is approximately 33,943 people per year.

Step 1: Identify the Initial and Final Values

The initial attendance in 1990 was 2,573,225, and the final attendance in 2004 was 3,048,427. These are the values we'll use to calculate the average annual rate of increase.

Step 2: Calculate the Change in Attendance

Subtract the initial attendance from the final attendance to find the total increase over the years. \[\text{Change in attendance} = 3,048,427 - 2,573,225 = 475,202\]

Step 3: Calculate the Number of Years

Subtract the initial year from the final year to find out over how many years this change occurred. \[\text{Number of years} = 2004 - 1990 = 14\]

Step 4: Calculate the Average Annual Rate of Increase

Divide the change in attendance by the number of years to find the average annual increase.\[\text{Average annual rate of increase} = \frac{475,202}{14} \approx 33,943\]

Step 5: Conclusion: Formulate the Final Answer

The average annual rate of increase in attendance is approximately 33,943 additional people per year.

Key Concepts

Problem Solving

Problem solving in mathematics often involves breaking down a complex task into smaller, manageable steps. By identifying the components of a problem, such as initial and final values or understanding what is being asked, you can create a clear path toward finding a solution. When dealing with the average rate of change, you need to translate word problems into mathematical calculations. It begins with recognizing what data is given and what is needed. Here, we need to determine the average annual rate of increase in attendance for the St. Louis Cardinals' games over 14 years. Once the problem is understood, it's all about solving it in a structured method. A systematic approach involves understanding the context, analyzing the values, performing calculations, and finally interpreting the results.

Step by Step Solution

Finding a solution step by step aids in comprehending complex ideas without feeling overwhelmed.

• Identify the Values: Start by pinpointing the initial and final attendance values: 2,573,225 in 1990 and 3,048,427 in 2004.
• Calculate the Change: Subtract to find the total increase in attendance: \(3,048,427 - 2,573,225 = 475,202\).
• Determine the Duration: Find the number of years between 1990 and 2004: \(2004 - 1990 = 14\) years.
• Average Annual Rate: Divide the change by the duration: \(\frac{475,202}{14} \approx 33,943\).

Each step builds on the last, ensuring clarity and correctness in the final solution.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules to manipulate these symbols. It allows us to express real-world problems in mathematical terms.In the given exercise, algebra concepts help us understand how to find an average rate of change using operations like subtraction and division. This exercise illustrates how algebra transforms a word problem into an equation like \( \frac{\text{Change in attendance}}{\text{Number of years}} \). By representing changes in values as equations, algebra allows for easy interpretation and solution of what might initially seem to be complicated problems.