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. $\quad 175(1.30)^t>1,200$
B. $175(1.03)^{ t }>1,200$
C. $175(0.30)^t<1,200$
D. $\quad 175(0.70)^t<1,200$
Answer (1)
The problem involves exponential growth, where the number of people in a concert hall increases by 30% per hour.
The initial number of people is 175, and the maximum occupancy is 1200.
The inequality representing when the number of people exceeds the occupancy limit is 1200"> 175 ( 1.30 ) t > 1200 .
Therefore, the correct answer is A. 1,200}"> 175 ( 1.30 ) t > 1 , 200
Explanation
Problem Analysis Let's analyze the problem. We are given that the maximum occupancy of a concert hall is 1,200 people. Initially, 175 people enter the hall. The number of people increases at a rate of 30% per hour. We need to find the inequality that represents the number of hours, t , after which the number of people in the hall exceeds the occupancy limit.
Exponential Growth Model The number of people in the hall after t hours can be modeled by an exponential growth function. The general form is: initial amount * (1 + growth rate)^time. In this case, the initial amount is 175, and the growth rate is 30% or 0.30. So, the number of people after t hours is 175 ( 1 + 0.30 ) t = 175 ( 1.30 ) t .
Setting up the Inequality We want to find the inequality that represents when the number of people exceeds 1,200. So, we set up the inequality: 1200"> 175 ( 1.30 ) t > 1200 .
Identifying the Correct Option Comparing this inequality with the given options, we see that option A matches our derived inequality.
Final Answer Therefore, the correct answer is A. 1,200"> 175 ( 1.30 ) t > 1 , 200 .
Examples
Exponential growth is a powerful tool for modeling real-world phenomena. For example, imagine you invest 1 , 000 ina s a v in g s a cco u n tt ha t e a r n s 5 t ye a rs , t h e am o u n t o f m o n ey in yo u r a cco u n t c anb e m o d e l e d b y t h ee q u a t i o n A = 1000(1.05)^t$. This equation helps you predict how your investment will grow over time, allowing you to plan for future financial goals. Understanding exponential growth is crucial in various fields, including finance, biology (population growth), and even computer science (algorithm complexity).
