Question

example 6 check attendance for division i college football bowl games can be modeled by 2069x + y = 52,648, where y is the average attendance and x is the number of years since 2010. estimate the attendance for division i college football bowl games in 2020. justify your estimate. is attendance expected to double? why? any constraints?

Explanation:

Step1: Calculate the value of x for 2020

The year 2020 is 2020 - 2010 = 10 years since 2010, so $x = 10$.

Step2: Substitute x into the equation

Substitute $x = 10$ into the equation $2069x + y=52648$. We get $2069\times10 + y=52648$.

Step3: Solve for y

First, calculate $2069\times10 = 20690$. Then the equation becomes $20690 + y=52648$. Subtract 20690 from both sides: $y=52648 - 20690=31958$.

Answer:

The estimated attendance for Division I college football bowl - games in 2020 is 31958. Justification: We used the linear model $2069x + y = 52648$, where $x$ represents the number of years since 2010. By substituting $x = 10$ (for the year 2020) into the equation and solving for $y$ (the average attendance), we obtained the estimate. Regarding whether the attendance is expected to double:
The initial attendance (when $x = 0$, i.e., in 2010) is $y=52648$ (since when $x = 0$, $2069\times0 + y=52648$). Double of 52648 is $2\times52648 = 105296$.
From the equation $y=52648 - 2069x$, as $x$ increases, $y$ decreases because of the negative coefficient of $x$. So the attendance is not expected to double. Constraints include the linear - nature of the model which may not hold in the long - run due to various factors like changes in popularity of the sport, economic conditions, competition from other events, etc.