Question

Seven days a year, Tiger Stadium becomes the fifth largest city in the state of Louisiana. Over 92,000 fans pack the stadium to watch the Tigers play. After the game, if the fans leave at a rate of 10% per minute, how long will it take before the stadium is half empty?

1. Find the data using at least 10 numbers in the x column.
2. Create a scatter plot. Label the graph and show increments.
3. Write an exponential equation.
4. Interpret the meaning of the "a" and "b" in your function y=ab^x including the units.
5. Find out how long it will take before the stadium is half empty and all the way empty.

Answer 1

1. To find the data, we can use the formula:

y = 92000(0.9)^x

where y is the number of fans remaining in the stadium after x minutes.

Using this formula, we can plug in x values from 0 to 30 (or more) to generate a table of values:

| x | y |
|------|---------|
| 0 | 92000 |
| 1 | 82800 |
| 2 | 74520 |
| 3 | 67068 |
| 4 | 60361 |
| 5 | 54325 |
| 6 | 48892 |
| 7 | 43903 |
| 8 | 39313 |
| 9 | 35082 |
| 10 | 31174 |

2. Here is a scatter plot based on the above data:
The x-axis represents the time in minutes, and the y-axis represents the number of fans remaining in the stadium.

3. The exponential equation that best fits the data is:

y = 92000(0.9)^x

where y is the number of fans remaining in the stadium after x minutes.

4. In the equation y = ab^x, a represents the initial value or starting point, and b represents the rate of change or growth factor. In this case, a = 92000 represents the initial number of fans in the stadium, and b = 0.9 represents the rate at which the number of fans decreases per minute. The units of a are fans, and the units of b are fans per minute.

5. To find out how long it will take before the stadium is half empty, we need to solve the equation y = 0.5a for x:

0.5a = 92000(0.9)^x

0.5(92000) = 92000(0.9)^x

0.5 = 0.9^x

Taking the logarithm of both sides, we get:

log(0.5) = x log(0.9)

x = log(0.5) / log(0.9)

x ≈ 7.72 minutes

Therefore, it will take approximately 7.72 minutes for the stadium to be half empty.

To find out how long it will take for the stadium to be completely empty, we need to solve the equation y = 0 for x:

0 = 92000(0.9)^x

Taking the logarithm of both sides, we get:

log(0) = x log(0.9)

This equation has no real solution, which means that the stadium will never be completely empty (in theory). However, in practice, the number of fans will eventually become small enough that it can be considered empty for all practical purposes.