A colony of bacteria is growing exponentially according to the function below, where [tex]$t$[/tex] is in hours. What will the approximate number of bacteria be after 6 hours?
[tex]$B(t)=4 \cdot e^{0.8 t}$[/tex]
A. 122
B. 486
C. 94,251
D. 8602
Answer (2)
Substitute t = 6 into the function B ( t ) = 4 b u ll e t e 0.8 t .
Calculate B ( 6 ) = 4 b u ll e t e 0.8 b u ll e t 6 = 4 b u ll e t e 4.8 .
Approximate the value of e 4.8 a pp ro x 121.5104 .
Multiply the result by 4 to find the approximate number of bacteria after 6 hours: B ( 6 ) a pp ro x 486 .
The approximate number of bacteria after 6 hours is 486 .
Explanation
Understanding the Problem We are given the function B ( t ) = 4 b u ll e t e 0.8 t that models the exponential growth of a bacteria colony, where t is measured in hours. We want to find the approximate number of bacteria after 6 hours, so we need to evaluate B ( 6 ) .
Substituting t=6 To find the number of bacteria after 6 hours, we substitute t = 6 into the function: B ( 6 ) = 4 b u ll e t e 0.8 b u ll e t 6 = 4 b u ll e t e 4.8 Now we need to calculate the value of e 4.8 and multiply it by 4.
Calculating B(6) Using a calculator, we find that e 4.8 a pp ro x 121.5104 . Therefore, B ( 6 ) = 4 b u ll e t 121.5104 a pp ro x 486.0416 Since we are looking for the approximate number of bacteria, we can round this to the nearest whole number.
Final Answer Rounding 486.0416 to the nearest whole number, we get 486. Therefore, the approximate number of bacteria after 6 hours is 486.
Examples
Exponential growth models, like the one in this problem, are used in various real-world scenarios. For example, they can model population growth, compound interest, and the spread of diseases. Understanding exponential growth helps in making predictions and informed decisions in fields like finance, biology, and public health. In finance, it helps to predict the growth of investments over time. In biology, it helps to understand how quickly a population of bacteria can grow under ideal conditions. In public health, it helps to model the spread of infectious diseases and implement effective control measures.
After substituting t = 6 into the function B ( t ) = 4 ⋅ e 0.8 t and calculating, the approximate number of bacteria after 6 hours is 486. Therefore, the answer is option B. 486.
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