The doubling period of a bacterial population is 20 minutes. At time [tex]$t=110$[/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)
The initial population is found using the formula P ( t ) = P 0 "." 2 20 t and the given population at t = 110 minutes.
Calculate P 0 = 2 5.5 90000 ≈ 1988.74 .
The population after 5 hours (300 minutes) is calculated using P ( 300 ) = P 0 "." 2 15 .
Calculate P ( 300 ) = 90000 ⋅ 2 9.5 ≈ 65166960.95 . The initial population is approximately 1988.74 and the population after 5 hours is approximately 65166960.95 .
Explanation
Setting up the model Let P ( t ) be the population at time t . Since the doubling period is 20 minutes, we can model the population growth as P ( t ) = P 0 ⋅ 2 20 t , where P 0 is the initial population at t = 0 . We are given that P ( 110 ) = 90000 . So, we have the equation:
Using the given information 90000 = P 0 "." 2 20 110 = P 0 "." 2 5.5
Finding the initial population To find the initial population P 0 , we solve for P 0 :
Calculating initial population P 0 = 2 5.5 90000 = 2 5.5 90000 ≈ 1988.74
Converting time to minutes Now, we want to find the population after 5 hours. First, convert 5 hours to minutes:
Calculating total minutes 5 hours × 60 hour minutes = 300 minutes
Finding the population after 5 hours We need to find P ( 300 ) . Using the formula P ( t ) = P 0 ⋅ 2 20 t , we have:
Substituting values P ( 300 ) = P 0 "." 2 20 300 = P 0 "." 2 15
Substituting initial population Substitute the value of P 0 we found earlier:
Calculating final population P ( 300 ) = 2 5.5 90000 ⋅ 2 15 = 90000 ⋅ 2 15 − 5.5 = 90000 ⋅ 2 9.5 ≈ 65166960.95
Final Answer Therefore, the initial population at time t = 0 is approximately 1988.74, and the population after 5 hours is approximately 65166960.95.
Examples
Bacterial growth models are used in various fields, such as medicine and environmental science. For example, in medicine, understanding the growth rate of bacteria helps determine the appropriate dosage of antibiotics. In environmental science, these models can predict the spread of harmful bacteria in water sources, aiding in the development of effective sanitation strategies. By understanding exponential growth, we can better manage and control bacterial populations in different scenarios.
The initial population at time t=0 was approximately 1987.24. After 5 hours, the bacterial population is expected to be around 65,536,000.
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