Question

The maximum occupancy of a concert hall is 1,200 people. The hall is hosting a concert, and 175 people enter as soon as the doors open in the morning. The number of people coming into the hall then increases at a rate of 30% per hour. If t represents the number of hours since the doors open, which inequality can be used to determine the number of hours after which the amount of people in the concert hall will exceed the occupancy limit?

Answer 1

To determine when the number of people in the concert hall will exceed the maximum occupancy of 1,200 people, we can set up an inequality based on the initial amount of people and the rate of increase.

Initially, when the doors open, there are 175 people in the hall. After that, the number of people increases by 30% per hour.

The formula for the number of people in the hall after \( t \) hours, given the initial amount and the growth rate, can be expressed as:

\[
N(t) = 175 \times (1.3)^t
\]

where \( N(t) \) is the number of people in the hall after \( t \) hours.

To find out when this number exceeds the maximum occupancy (1,200), we set up the inequality:

\[
175 \times (1.3)^t > 1200
\]

This inequality can be used to determine the number of hours \( t \) after which the number of people in the concert hall will exceed the occupancy limit of 1,200 people.

Therefore, the inequality you can use is:

\[
175 \times (1.3)^t > 1200
\]