Select the correct answer.
The number of cars that passed through a tollbooth prior to $6 a.m.$ is $1,380$. The number of cars that pass through the tollbooth from 6 a.m. through the morning rush hour increases by $46 \%$ every hour.
Which of the following inequalities can be used to determine the number of hours, $t$, after 6 a.m. when the number of cars that have passed through the tollbooth is over 4,300 ?
$1,380(1.46)^t<4,300$
$1,380(0.54)^t<4,300$
$1,380(1.46)^t>4,300$
$1,380(0.54)^t>4,300$
Answer (2)
The inequality that can be used to determine the number of hours, t , after 6 a.m. when the number of cars that have passed through the tollbooth exceeds 4,300 is 4,300"> 1 , 380 ( 1.46 ) t > 4 , 300 .
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The number of cars increases by 46% every hour, so we use the formula for exponential growth.
The formula is ini t ia l u mb e r o f c a rs ( 1 + g ro wt h _ r a t e ) t .
The initial number of cars is 1380, and the growth rate is 0.46, so the number of cars after t hours is 1380 ( 1.46 ) t .
The inequality that represents when the total number of cars is over 4300 is 4,300}"> 1 , 380 ( 1.46 ) t > 4 , 300 .
Explanation
Problem Analysis Let's analyze the problem. We are given that the number of cars that passed through a tollbooth prior to 6 a.m. is 1,380. The number of cars that pass through the tollbooth from 6 a.m. increases by 46% every hour. We want to find the inequality that can be used to determine the number of hours, t, after 6 a.m. when the number of cars that have passed through the tollbooth is over 4,300.
Setting up the Inequality The number of cars after t hours can be modeled by an exponential growth function. The formula for exponential growth is ini t ia l u mb e r o f c a rs u mb e r o f c a rs ( 1 + g ro wt h _ r a t e ) t . In this case, the initial number of cars is 1380, and the growth rate is 46% or 0.46. So, the number of cars after t hours is 1380 ( 1 + 0.46 ) t = 1380 ( 1.46 ) t . We want to find the inequality that represents when the total number of cars is over 4300. Therefore, the inequality is 4300"> 1380 ( 1.46 ) t > 4300 .
Final Answer The correct inequality is 4,300"> 1 , 380 ( 1.46 ) t > 4 , 300 .
Examples
Exponential growth is a concept that applies to many real-world situations, such as population growth, compound interest, and the spread of information. For example, if a city's population grows at a rate of 5% per year, we can use an exponential growth model to predict the population in the future. Similarly, if you invest money in a savings account that earns compound interest, the amount of money in your account will grow exponentially over time. Understanding exponential growth can help you make informed decisions in various aspects of your life.
