The doubling period of a bacterial population is 20 minutes. At time [tex]$t=90$[/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 5 hours.
Answer (2)
Establish the exponential growth formula: P ( t ) = P 0 ⋅ 2 T t , where T = 20 minutes.
Calculate the initial population P 0 using P ( 90 ) = 90000 : P 0 = 2 4.5 90000 ≈ 3977 .
Determine the population after 5 hours (300 minutes): P ( 300 ) = P 0 ⋅ 2 15 .
Find the population after 5 hours: P ( 300 ) ≈ 130333922 .
Explanation
Understanding the Problem We are given that the doubling period of a bacterial population is 20 minutes. This means that the population doubles every 20 minutes. We are also given that at time t = 90 minutes, the population is 90000. We need to find the initial population at time t = 0 and the population after 5 hours (300 minutes).
Setting up the Equation Let P ( t ) be the population at time t (in minutes). The general formula for exponential growth is given by P ( t ) = P 0 ⋅ 2 T t where P 0 is the initial population and T is the doubling time. In this case, T = 20 minutes.
Using the Given Information We know that P ( 90 ) = 90000 . Plugging this into the formula, we get 90000 = P 0 ⋅ 2 20 90 90000 = P 0 ⋅ 2 4.5
Calculating the Initial Population Now, we solve for P 0 :
P 0 = 2 4.5 90000 P 0 = 2 4.5 90000 ≈ 3977.48
Finding the Population After 5 Hours Next, we want to find the population after 5 hours, which is 300 minutes. We use the formula P ( 300 ) = P 0 ⋅ 2 20 300 P ( 300 ) = P 0 ⋅ 2 15
Calculating the Population After 5 Hours Substituting the value of P 0 we found earlier: P ( 300 ) = 2 4.5 90000 ⋅ 2 15 P ( 300 ) = 90000 ⋅ 2 15 − 4.5 P ( 300 ) = 90000 ⋅ 2 10.5 P ( 300 ) ≈ 130333921.91
Final Answer Therefore, the initial population at time t = 0 is approximately 3977, and the population after 5 hours is approximately 130,333,922.
Examples
Bacterial growth models are crucial in various real-world applications. For instance, in food safety, understanding the growth rate of bacteria helps determine the shelf life of products and prevent spoilage. In medicine, these models aid in predicting the spread of infections and optimizing antibiotic dosages. Moreover, in biotechnology, controlling bacterial growth is essential for producing valuable substances like enzymes or biofuels. By applying mathematical models, we can effectively manage and utilize bacterial populations in diverse fields.
The initial bacterial population at time t = 0 is approximately 3977, and the population after 5 hours is approximately 130,333,922.
;
