Question

There are currently 100,000 people in a stadium watching a soccer game. when the game ends, about 3% of the crowd will leave the stadium each minute. at this rate, how many minutes will it take for 3/4 of the crowd to leave the stadium?

m = num of min it takes for 3/4 of crowd to leave stadium
t(m) = num of people in stadium after m min
a = the num of people currently in stadium
r = percent change in number of people in stadium
t(m) = a(1+r)^m

Answer 1

Use the exponential decay formula to determine when 1/4 of the initial stadium crowd remains, by first calculating 1/4 of the original number of people and then solving for the time using logarithms.

Step-by-step explanation:

To answer the question about how long it will take for 3/4 of the crowd to leave the stadium, we shall use the given exponential decay model t(m) = a(1+r)^m, where:

• a is the initial number of people in the stadium (100,000 people).
• r is the rate of change (-0.03, since 3% are leaving).
• t(m) is the number of people after m minutes.
• m is the number of minutes it takes for 3/4 of the crowd to leave.

To find when 3/4 of the crowd will have left, we need to solve for the time when the number of people left in the stadium is 1/4 of the initial amount:

1. Calculate 1/4 of the original crowd: 1/4 of 100,000 people is 25,000.
2. Substitute into our equation: 25,000 = 100,000(1-0.03)^m.
3. Solve for m by using logarithms.

The final answer will give us the number of minutes it takes for the crowd to reduce to 25,000 people.