Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
To solve this problem, we need to follow a step-by-step approach using the given information about the doubling period and the bacterial population.
### Part 1: Finding the Initial Population at Time [tex]\( t = 0 \)[/tex]
1. Given Information:
- Doubling period [tex]\( T_d = 10 \)[/tex] minutes
- Population at [tex]\( t = 100 \)[/tex] minutes is [tex]\( P_{100} = 60000 \)[/tex]
2. Determine the number of doublings between time [tex]\( t = 0 \)[/tex] and [tex]\( t = 100 \)[/tex] minutes:
[tex]\[ \text{Number of doublings} = \frac{\text{Total time}}{\text{Doubling period}} = \frac{100 \text{ minutes}}{10 \text{ minutes}} = 10 \][/tex]
3. Finding the initial population [tex]\( P_0 \)[/tex]:
We know that the population doubles 10 times from [tex]\( t = 0 \)[/tex] to [tex]\( t = 100 \)[/tex] minutes. Therefore, the relationship between the initial population [tex]\( P_0 \)[/tex] and the population at [tex]\( t = 100 \)[/tex] minutes is:
[tex]\[ P_{100} = P_0 \times 2^{\text{Number of doublings}} \][/tex]
Substituting the known values:
[tex]\[ 60000 = P_0 \times 2^{10} \][/tex]
Solving for [tex]\( P_0 \)[/tex]:
[tex]\[ P_0 = \frac{60000}{2^{10}} = \frac{60000}{1024} \approx 58.59375 \][/tex]
So, the initial population at time [tex]\( t = 0 \)[/tex] was approximately [tex]\( 58.59375 \)[/tex].
### Part 2: Finding the Population After 4 Hours
4. Given Information:
- 4 hours = 240 minutes
5. Determine the number of doublings in 240 minutes:
[tex]\[ \text{Number of doublings} = \frac{240 \text{ minutes}}{10 \text{ minutes}} = 24 \][/tex]
6. Finding the population after 4 hours:
We now use the initial population [tex]\( P_0 \)[/tex] found in Part 1 to calculate the population after 4 hours. The relationship between the initial population and the population after 240 minutes is:
[tex]\[ P_{240} = P_0 \times 2^{\text{Number of doublings}} \][/tex]
Substituting the known values:
[tex]\[ P_{240} = 58.59375 \times 2^{24} \][/tex]
Since [tex]\( 2^{24} = 16,777,216 \)[/tex]:
[tex]\[ P_{240} = 58.59375 \times 16,777,216 = 983,040,000 \][/tex]
So, the bacterial population after 4 hours is [tex]\( 983,040,000 \)[/tex].
### Final Answers:
- The initial population at time [tex]\( t = 0 \)[/tex] was approximately [tex]\( 58.59375 \)[/tex].
- The population after 4 hours (240 minutes) will be [tex]\( 983,040,000 \)[/tex].
### Part 1: Finding the Initial Population at Time [tex]\( t = 0 \)[/tex]
1. Given Information:
- Doubling period [tex]\( T_d = 10 \)[/tex] minutes
- Population at [tex]\( t = 100 \)[/tex] minutes is [tex]\( P_{100} = 60000 \)[/tex]
2. Determine the number of doublings between time [tex]\( t = 0 \)[/tex] and [tex]\( t = 100 \)[/tex] minutes:
[tex]\[ \text{Number of doublings} = \frac{\text{Total time}}{\text{Doubling period}} = \frac{100 \text{ minutes}}{10 \text{ minutes}} = 10 \][/tex]
3. Finding the initial population [tex]\( P_0 \)[/tex]:
We know that the population doubles 10 times from [tex]\( t = 0 \)[/tex] to [tex]\( t = 100 \)[/tex] minutes. Therefore, the relationship between the initial population [tex]\( P_0 \)[/tex] and the population at [tex]\( t = 100 \)[/tex] minutes is:
[tex]\[ P_{100} = P_0 \times 2^{\text{Number of doublings}} \][/tex]
Substituting the known values:
[tex]\[ 60000 = P_0 \times 2^{10} \][/tex]
Solving for [tex]\( P_0 \)[/tex]:
[tex]\[ P_0 = \frac{60000}{2^{10}} = \frac{60000}{1024} \approx 58.59375 \][/tex]
So, the initial population at time [tex]\( t = 0 \)[/tex] was approximately [tex]\( 58.59375 \)[/tex].
### Part 2: Finding the Population After 4 Hours
4. Given Information:
- 4 hours = 240 minutes
5. Determine the number of doublings in 240 minutes:
[tex]\[ \text{Number of doublings} = \frac{240 \text{ minutes}}{10 \text{ minutes}} = 24 \][/tex]
6. Finding the population after 4 hours:
We now use the initial population [tex]\( P_0 \)[/tex] found in Part 1 to calculate the population after 4 hours. The relationship between the initial population and the population after 240 minutes is:
[tex]\[ P_{240} = P_0 \times 2^{\text{Number of doublings}} \][/tex]
Substituting the known values:
[tex]\[ P_{240} = 58.59375 \times 2^{24} \][/tex]
Since [tex]\( 2^{24} = 16,777,216 \)[/tex]:
[tex]\[ P_{240} = 58.59375 \times 16,777,216 = 983,040,000 \][/tex]
So, the bacterial population after 4 hours is [tex]\( 983,040,000 \)[/tex].
### Final Answers:
- The initial population at time [tex]\( t = 0 \)[/tex] was approximately [tex]\( 58.59375 \)[/tex].
- The population after 4 hours (240 minutes) will be [tex]\( 983,040,000 \)[/tex].
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.