Problem 17 - Applied Calculus

Question

Table \(1.13\) shows attendance at NFL football games. (a) Find the average rate of change in the attendance from 2003 to 2007 . Give units. (b) Find the annual increase in the attendance for each year from 2003 to 2007 . (Your answer should be four numbers.) (c) Show that the average rate of change found in part (a) is the average of the four yearly changes found in part (b). $$ \begin{array}{c|c|c|c|c|c} \hline \text { Year } & 2003 & 2004 & 2005 & 2006 & 2007 \\ \hline \text { Attendance } & 21.64 & 21.71 & 21.79 & 22.20 & 22.26 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

(a) 0.155 million attendees/year; (b) 0.07, 0.08, 0.41, 0.06 million; (c) Verified.

Step 1: Understand the Problem

First, we need to determine what is being asked. Part (a) asks for the average rate of change of attendance from 2003 to 2007. Part (b) asks for the annual change each year, and part (c) asks us to relate these two sets of information.

Step 2: Calculate Average Rate of Change (2003-2007)

The average rate of change from 2003 to 2007 is found by taking the change in attendance divided by the change in years. Given: - Attendance in 2003: 21.64 - Attendance in 2007: 22.26 - Number of years from 2003 to 2007: 4 Average rate of change = \( \frac{22.26 - 21.64}{2007 - 2003} = \frac{22.26 - 21.64}{4} = \frac{0.62}{4} = 0.155 \text{ million attendees per year} \).

Step 3: Calculate Annual Increase for Each Year

Calculate the increase in attendance for each year: - 2004 - 2003: \( 21.71 - 21.64 = 0.07 \text{ million} \)- 2005 - 2004: \( 21.79 - 21.71 = 0.08 \text{ million} \)- 2006 - 2005: \( 22.20 - 21.79 = 0.41 \text{ million} \)- 2007 - 2006: \( 22.26 - 22.20 = 0.06 \text{ million} \).

Step 4: Verify Average Rate (Part a vs. Part b)

Calculate the average of the annual changes found in part (b): Average = \( \frac{0.07 + 0.08 + 0.41 + 0.06}{4} = \frac{0.62}{4} = 0.155 \text{ million attendees per year} \).Since the result matches part (a), this confirms the average rate of change is consistent with the yearly changes.

Key Concepts

Attendance Analysis

The attendance analysis involves examining the number of people attending NFL football games over several years. By analyzing this data, we can discover trends, such as increases or decreases in attendance over time. This process helps us identify patterns and make predictions about future attendance. For example, a consistent increase in attendance could suggest growing popularity, while a decrease might indicate issues that need addressing.

• Analyzing data: The table presents attendance from 2003 to 2007.
• Comparing numbers: Attendance numbers are given in millions for each year.
• Observing changes: We need to look at these numbers year by year to understand trends.

By carefully examining these yearly figures, we can grasp how changes in attendance reflect broader trends within the NFL.

Rate of Change Calculation

To calculate the average rate of change, we need to demonstrate how attendance numbers have shifted over time. This involves comparing attendance statistics between two points in time and dividing the difference in attendance by the number of years between those points. It's a valuable mathematical concept that helps describe the rate at which attendance grows:

• Define the time interval: From 2003 to 2007 (4 years).
• Measure the change: Subtract the initial attendance (2003) from the final attendance (2007).
• Convert to rate: Divide this change by the number of years between these two dates.

For example, the attendance grew from 21.64 million in 2003 to 22.26 million in 2007, resulting in an average rate of change of 0.155 million attendees per year, indicating steady growth in attendance over this period.

Data Interpretation

Data interpretation is key to understanding what the calculated rate of change implies. This allows us to form conclusions about the trends shown by the data. In this context, evaluating the annual change in attendance helps us see how each year's growth contributes to the larger trend. This involves looking at each separate year's increase and how those numbers align with the average calculated:

• Calculate annual increases: For instance, increases are 0.07 million in 2004 and 0.41 million in 2006.
• Link increases: Connect each year's growth to the overall trend seen between 2003 and 2007.
• Confirm hypotheses: Ensure the sum of these yearly increments matches the average rate previously calculated.

By analyzing the data closely, we can verify that the overall average growth matches these separate yearly improvements, which further interprets the trend as consistent and reliable.

Annual Growth

Annual growth signifies the year-over-year increase in attendance. By examining year-to-year figures, we can comprehend how attendance modifies annually. Each improvement represents a shift in how many individuals decide to attend football games, which is critical for understanding both short-term changes and long-term trends:

• Annual observations: Increment from 0.07 million in 2004 to 0.41 million in 2006 is significant.
• Identify variations: Noticeable spikes occur, such as the sharp increase in 2006.
• Evaluate influences: Determine what causes these changes, like increased fan interest or team performance.

Continuous recording and analysis of annual growth help in planning and forecasting future attendance patterns. These insights enable organizations, such as the NFL, to make informed decisions to sustain or boost attendance.