Discover a wealth of knowledge at Westonci.ca, where experts provide answers to your most pressing questions. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Discover in-depth answers to your questions from a wide network of professionals on our user-friendly Q&A platform.
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.
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.
Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for visiting Westonci.ca, your go-to source for reliable answers. Come back soon for more expert insights.