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Select the correct answer.

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 entering the hall then increases at a rate of [tex]$30\%$[/tex] per hour. If [tex]f[/tex] represents the number of hours since the doors opened, which inequality can be used to determine the number of hours after which the number of people in the concert hall will exceed the occupancy limit?

A. [tex]175(1.30)^t > 1,200[/tex]
B. [tex]175(1.08)^t > 1,200[/tex]
C. [tex]175(0.70)^t < 1,200[/tex]
D. [tex]175(0.30)^t < 1,200[/tex]

Answer :

We are given:

- The maximum occupancy is [tex]$1,\!200$[/tex] people.
- Initially, [tex]$175$[/tex] people enter the hall.
- The number of people increases by [tex]$30\%$[/tex] per hour.

Since increasing by [tex]$30\%$[/tex] per hour means multiplying by [tex]$1.30$[/tex] every hour, after [tex]$t$[/tex] hours the number of people in the hall is given by

[tex]$$
175 \cdot (1.30)^t.
$$[/tex]

To find when the number of people exceeds the maximum occupancy, we set up the inequality:

[tex]$$
175 \cdot (1.30)^t > 1200.
$$[/tex]

This inequality is used to determine the number of hours after which the concert hall occupancy exceeds [tex]$1,\!200$[/tex] people.

Thus, the correct answer is Option A.

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