The doubling period of a bacterial population is 20 minutes. At time [tex]$t=120$[/tex] minutes, the bacterial population was 90000. What was the initial population at time [tex]$t=0$[/tex]? Find the size of the bacterial population after 4 hours.
Answer (1)
The bacterial population growth is modeled as P ( t ) = P 0 ⋅ 2 ( t /20 ) .
Use the given data P ( 120 ) = 90000 to find the initial population P 0 = 2 6 90000 = 1406.25 .
Calculate the population after 4 hours (240 minutes) using P ( 240 ) = 1406.25 ⋅ 2 ( 240/20 ) = 5760000 .
The initial population is 1406.25 and the population after 4 hours is 5760000 .
Explanation
Understanding the Problem The problem states that a bacterial population doubles every 20 minutes. We know the population at time t = 120 minutes is 90000, and we want to find the initial population at t = 0 and the population after 4 hours (240 minutes).
Setting up the Model We can model the population growth with the formula P ( t ) = P 0 ⋅ 2 ( t /20 ) , where P ( t ) is the population at time t , P 0 is the initial population, and t is the time in minutes.
Using the Given Information We are given that P ( 120 ) = 90000 . Plugging this into our formula, we get:
90000 = P 0 ⋅ 2 ( 120/20 ) = P 0 ⋅ 2 6 = P 0 ⋅ 64
Calculating the Initial Population To find the initial population P 0 , we solve for P 0 :
P 0 = 64 90000 = 1406.25
Since we are dealing with a population, it makes sense to round to the nearest whole number. However, we will keep the decimal value for further calculations to maintain accuracy.
Calculating the Population After 4 Hours Now, we want to find the population after 4 hours, which is 240 minutes. We use the formula P ( t ) = P 0 ⋅ 2 ( t /20 ) with t = 240 :
P ( 240 ) = 1406.25 ⋅ 2 ( 240/20 ) = 1406.25 ⋅ 2 12 = 1406.25 ⋅ 4096 = 5760000
Final Answer Therefore, the initial population at time t = 0 was 1406.25, and the population after 4 hours is 5760000.
Examples
Bacterial growth is a classic example of exponential growth, which has many real-world applications. For instance, understanding bacterial growth rates is crucial in food safety to prevent spoilage and in medicine to control infections. Similarly, exponential growth models are used in finance to calculate compound interest and in environmental science to study population dynamics. By understanding the doubling period and applying the exponential growth formula, we can predict population sizes at different times, which is essential for making informed decisions in various fields.
