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 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?
A. 175(130)^t > 1200
B. 175(0.70)^t < 1200
C. 175(0.30)^t < 1200
D. 175(1.03)^t > 1200
The correct answer is A. 175(130)^t > 1200.
The number of people coming into the hall is increasing at a rate of 30% per hour. This means that the number of people at any given time t can be represented by 175(1.30)^t, since each hour the number of people is multiplied by 1.30.
To determine the number of hours after which the amount of people in the concert hall will exceed the occupancy limit of 1200, we can set up the following inequality:
175(1.30)^t > 1200
Simplifying the inequality gives us:
(1.30)^t > 1200/175
Taking the natural logarithm (ln) of both sides of the inequality allows us to solve for t:
t ln(1.30) > ln(1200/175)
t > ln(1200/175) / ln(1.30)
Therefore, the inequality 175(130)^t > 1200 can be used to determine the number of hours after which the amount of people in the concert hall will exceed the occupancy limit.