Question

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(0.30)^t < 1,200 b. 175(1.03)^t > 1,200 c. 175(0.70)^t < 1,200 d. 175(1.30)^t > 1,200

Explanation:

Step1: Identify the growth - model formula

The initial number of people is 175 and the growth rate is 30% or 0.3 per hour. The formula for exponential growth is $N = N_0(1 + r)^t$, where $N_0$ is the initial amount, $r$ is the growth rate, and $t$ is the time. Here, $N_0=175$, $r = 0.3$, so the number of people in the hall after $t$ hours is $175(1 + 0.3)^t=175(1.3)^t$.

Step2: Set up the inequality

We want to find the number of hours $t$ after which the number of people in the hall exceeds the occupancy limit of 1200. So the inequality is $175(1.3)^t>1200$.

Answer:

D. $175(1.30)^t>1200$