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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, [tex]$40\%$[/tex] of the sample remains alive. Which best describes the graph of the function that represents the number of live bacteria after [tex]$x$[/tex] 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]

Sagot :

To solve this problem, let's break it down step-by-step.

1. Initial Population of Bacteria:
- The initial number of bacteria before treatment is given as 5,000.

2. Daily Decay Rate:
- After each day of treatment, 40% of the bacteria remain alive. This implies that [tex]\(0.4\)[/tex] (or 40%) of the bacteria survive each day.

3. Decay Function:
- To find the number of bacteria remaining after [tex]\(x\)[/tex] days of treatment, we need to set up an exponential decay function. The general form of the exponential decay function is:
[tex]\[ f(x) = A \cdot (r)^x \][/tex]
where:
- [tex]\(A\)[/tex] is the initial quantity (in this case, 5,000),
- [tex]\(r\)[/tex] is the decay rate (in this case, 0.4),
- [tex]\(x\)[/tex] is the time in days.

4. Forming the Function:
- Substituting the given values into the formula, we get the function:
[tex]\[ f(x) = 5000 \cdot (0.4)^x \][/tex]

5. Analyzing the Asymptote:
- As [tex]\(x\)[/tex] (the number of days) increases, [tex]\((0.4)^x\)[/tex] gets smaller and smaller, approaching 0. Thus, the function [tex]\(f(x) = 5000 \cdot (0.4)^x\)[/tex] gets closer and closer to 0, but never actually reaches 0.
- This means that the graph of the function has a horizontal asymptote at [tex]\(y = 0\)[/tex].

6. Conclusion:
- Given the decay rate and the asymptote, the best description of the graph of the function is:
[tex]\[ f(x) = 5000 \cdot (0.4)^x, \text{ with a horizontal asymptote of } y = 0 \][/tex]

Hence, the correct description is:
[tex]\[ f(x) = 5000(0.4)^x, \text{ with a horizontal asymptote of } y = 0 \][/tex]
This matches the first option provided in the question.
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