A population of bacteria is treated with an antibiotic. It is estimated that 5,000 live bacteria existed in the sample before treatment. After each day of treatment, $40 \%$ of the sample remains alive. Which best describes the graph of the function that represents the number of live bacteria after $x$ days of treatment?
A. [tex]f(x)=5000(0.4)^x[/tex], with a horizontal asymptote of [tex]y=0[/tex]
B. [tex]f(x)=5000(0.6)^x[/tex], with a vertical asymptote of [tex]x =0[/tex]
C. [tex]f(x)=5000(1.4)^x[/tex], with a horizontal asymptote of [tex]y=0[/tex]
D. [tex]f(x)=5000(1.6)^x[/tex], with a vertical asymptote of [tex]x=0[/tex]
Answer (2)
The problem describes an exponential decay situation where a bacteria population decreases over time.
The function representing the number of live bacteria after x days is f ( x ) = 5000 ( 0.4 ) x .
The function has a horizontal asymptote at y = 0 , indicating the bacteria population approaches zero over time.
Therefore, the correct answer is f ( x ) = 5000 ( 0.4 ) x , with a horizontal asymptote of y = 0 .
Explanation
Understanding the Problem We are given that a population of bacteria starts with 5,000 live bacteria. After each day, 40% of the bacteria remain alive. We want to determine the function that represents the number of live bacteria after x days and identify its asymptote.
Setting up the Exponential Function The number of live bacteria after x days can be modeled by an exponential decay function of the form f ( x ) = A ( r ) x , where A is the initial number of bacteria and r is the decay factor. In this case, A = 5000 since we start with 5,000 bacteria.
Determining the Decay Factor Since 40% of the bacteria remain alive after each day, the decay factor is r = 0.4 . Therefore, the function is f ( x ) = 5000 ( 0.4 ) x .
Finding the Asymptote As x approaches infinity, ( 0.4 ) x approaches 0, so the function has a horizontal asymptote at y = 0 . This means that as time goes on, the number of live bacteria approaches 0.
Conclusion Comparing the derived function and asymptote with the given options, we see that the correct answer is f ( x ) = 5000 ( 0.4 ) x , with a horizontal asymptote of y = 0 .
Examples
Exponential decay is a concept used in various real-world scenarios. For example, it can model the depreciation of a car's value over time. If a car initially costs 25 , 000 an d l oses 15 x ye a rsc anb e m o d e l e d b y t h e f u n c t i o n V(x) = 25000(0.85)^x$. This helps in understanding how assets lose value and in making informed financial decisions.
The number of live bacteria after x days of antibiotic treatment can be represented by the function f ( x ) = 5000 ( 0.4 ) x , which indicates exponential decay. It has a horizontal asymptote at y = 0 , meaning the bacteria population approaches zero over time. Therefore, the correct choice is option A.
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