Suppose that the number of bacteria in a certain population increases according to a continuous exponential growth model, with a growth rate parameter of 16% per hour. Suppose also that a sample culture of 2900 bacteria is obtained from this population. Find the size of the sample after five hours. Round your answer to the nearest integer.
Note: This is a continuous exponential growth model. And though the growth rate parameter is 16% per hour, the actual growth is not 16% each hour.
Answer (2)
Use the continuous exponential growth model: N ( t ) = N 0 e k t .
Substitute the given values: N 0 = 2900 , k = 0.16 , and t = 5 .
Calculate N ( 5 ) = 2900 e 0.16 × 5 = 2900 e 0.8 ≈ 6454.068692628157 .
Round the result to the nearest integer: 6454 .
Explanation
Problem Setup We are given that the number of bacteria increases according to a continuous exponential growth model with a growth rate parameter of 16% per hour. We are also given that the initial sample size is 2900 bacteria. We want to find the size of the sample after 5 hours.
Exponential Growth Formula The continuous exponential growth model is given by the formula:
N ( t ) = N 0 e k t
where:
N ( t ) is the number of bacteria at time t
N 0 is the initial number of bacteria
k is the growth rate parameter
t is the time in hours
Given Values In this problem, we have:
N 0 = 2900
k = 0.16
t = 5
We want to find N ( 5 ) .
Plugging in the values Plugging in the values, we get:
N ( 5 ) = 2900 e 0.16 × 5 = 2900 e 0.8
Calculating the final number We know that e 0.8 ≈ 2.225540928492468 , so
N ( 5 ) = 2900 × 2.225540928492468 ≈ 6454.068692628157
Rounding the result Rounding to the nearest integer, we get N ( 5 ) ≈ 6454 .
Final Answer Therefore, the size of the sample after 5 hours is approximately 6454 bacteria.
Examples
Exponential growth models are used in various real-world scenarios, such as calculating population growth, compound interest, and the spread of infectious diseases. For instance, epidemiologists use these models to predict the number of infected individuals during an outbreak, which helps in planning and implementing effective control measures. Similarly, financial analysts use exponential growth models to estimate the future value of investments, aiding in making informed financial decisions. Understanding exponential growth is crucial in many fields for predicting and managing growth-related phenomena.
Using the continuous exponential growth model, the size of the bacteria population after five hours, starting from an initial sample of 2900 bacteria with a growth rate of 16% per hour, is about 6454 bacteria. This is calculated using the formula N ( t ) = N 0 e k t and rounding the result to the nearest integer. Thus, after 5 hours, you can expect to see approximately 6454 bacteria.
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