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? f(x) = 5000(0.4)x, with a horizontal asymptote of y = 0 f(x) = 5000(0.6)x, with a vertical asymptote of x = 0 f(x) = 5000(1.4)x, with a horizontal asymptote of y = 0 f(x) = 5000(1.6)x, with a vertical asymptote of x = 0
The Correct Answer and Explanation is:
The correct answer is:
f(x) = 5000(0.4)^x, with a horizontal asymptote of y = 0.
Explanation:
This problem describes a situation where a population of bacteria is being treated with an antibiotic. The population decreases over time, and after each day, 40% of the remaining bacteria survive. This situation suggests a decaying exponential function.
Let’s break down the components:
- Initial Population: The number of live bacteria initially is 5,000. This is the starting value, which is represented as 5000 in the equation.
- Rate of Decay: The problem states that 40% of the population remains alive after each day of treatment. This means the bacteria population reduces by 60% each day (since 100% – 40% = 60%). To represent this mathematically, the multiplier is 0.4, as 40% of the population remains each day.
- Exponential Decay: The general form for exponential decay is:
[
f(x) = P \cdot (r)^x
]
where:
- (P) is the initial population (5,000),
- (r) is the rate of decay (0.4, representing the remaining population after each day),
- (x) is the number of days (time variable).
Thus, the equation that represents the bacteria population after (x) days is:
[
f(x) = 5000 \cdot (0.4)^x
]
This shows that each day, the population decreases by a factor of 0.4.
- Horizontal Asymptote: As time increases, the number of bacteria approaches zero. This suggests that the population will never be exactly zero, but it will get infinitely close to it. This behavior is represented by a horizontal asymptote at (y = 0). The horizontal asymptote occurs because the function approaches zero but never crosses it.
Why the Other Options are Incorrect:
- f(x) = 5000(0.6)^x, with a vertical asymptote of x = 0: The factor 0.6 would imply that 60% of the population remains, which contradicts the problem stating that 40% remains alive. Additionally, a vertical asymptote at (x = 0) is incorrect because the function is defined at (x = 0) and has no vertical asymptote.
- f(x) = 5000(1.4)^x, with a horizontal asymptote of y = 0: This would represent exponential growth (1.4 is greater than 1), not decay. Since the population is decreasing, this is not the correct function.
- f(x) = 5000(1.6)^x, with a vertical asymptote of x = 0: Again, 1.6 implies growth, not decay, and the vertical asymptote at (x = 0) is incorrect for the same reasons as the previous one.
Therefore, the correct function is f(x) = 5000(0.4)^x, with a horizontal asymptote of y = 0.