Answered

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$175(0.70)^t < 1,200$
C$175(1.03)^t > 1,200$
D$175(1.30)^t > 1,200$

Verified Answer :

  • Identify the initial population \(P_0 = 175\) and the growth rate \(r = 30\%\) per hour.
  • Express the population at time \(t\) as the function \(P(t) = 175(1.30)^t\).
  • Set the condition for exceeding the capacity limit of \(1,200\) as \(P(t) > 1,200\).
  • The resulting inequality is \(\boxed{175(1.30)^t > 1,200}\).

Explanation

  1. Analyzing the Growth Function

The problem asks us to determine an inequality that models the population growth in a concert hall to find when the number of people exceeds the occupancy limit.
Given data:

  • The initial number of people at \(t = 0\) is \(175\).
  • The growth rate is \(30\%\) per hour.
  • The maximum occupancy (limit) is \(1,200\).
  • \(t\) represents the number of hours since the doors opened.

We know that an initial amount \(P_0\) growing at a rate of \(r\%\) per time period is modeled by the exponential growth function \(P(t) = P_0(1 + \frac{r}{100})^t\).
Here, \(P_0 = 175\) and \(r = 30\), so the growth factor is \(1 + \frac{30}{100} = 1.30\).
Thus, the number of people in the hall after \(t\) hours is given by the expression \(175(1.30)^t\).

  1. Formulating the Inequality

We want to find the number of hours \(t\) after which the total number of people in the concert hall exceeds the maximum capacity of \(1,200\).
The condition "exceeds" corresponds to the mathematical operator \(>\).
Setting the population function to exceed the capacity limit, we write the inequality:
\[175(1.30)^t > 1,200\]
This inequality correctly captures the initial state, the hourly percentage growth, and the constraint imposed by the concert hall's capacity. Comparing this with the provided choices, it matches the fourth option.

  1. Conclusion

The correct inequality that represents the scenario is \(175(1.30)^t > 1,200\). This represents the exponential growth of the crowd size over time, indicating when the occupancy limit of \(1,200\) will be surpassed. The final answer is \(\boxed{175(1.30)^t > 1,200}\).

Examples

Exponential growth models are widely used in various fields to predict future outcomes based on current trends. For instance, in biology, they model the growth of a bacterial population; in finance, they calculate compound interest on an investment; and in city planning, they help estimate the future population of a growing urban area to ensure that resources like housing and utilities meet the demand. By mastering these equations, you can better understand and plan for systems that grow or decay over time.

Related Questions