Get the answers you've been looking for with the help of IDNLearn.com's expert community. Discover in-depth answers to your questions from our community of experienced professionals.
Sagot :
To solve this problem, we need to use the information about the bacterial growth, which doubles every 20 minutes. Here’s a detailed, step-by-step solution:
### Step 1: Finding the Initial Population at Time [tex]\( t = 0 \)[/tex]
1. Identify the Known Values:
- Doubling period: 20 minutes
- Given time (t): 100 minutes
- Bacterial population at [tex]\( t = 100 \)[/tex] minutes: 60,000
2. Calculate the Number of Doublings from [tex]\( t = 0 \)[/tex] to [tex]\( t = 100 \)[/tex]:
Since the doubling period is 20 minutes, the number of doublings that occur in 100 minutes is calculated as:
[tex]\[ \text{Number of doublings} = \frac{100 \text{ minutes}}{20 \text{ minutes/doubling}} = 5 \][/tex]
3. Determine the Initial Population Using the Doubling Formula:
The relationship between the initial population (P0) and the population at time [tex]\( t = 100 \)[/tex] minutes (P100) can be described by the formula:
[tex]\[ P_{100} = P_0 \times (2^{\text{Number of doublings}}) \][/tex]
Rearranging this formula to solve for [tex]\( P_0 \)[/tex]:
[tex]\[ P_0 = \frac{P_{100}}{2^{\text{Number of doublings}}} \][/tex]
4. Plug in the Known Values:
[tex]\[ P_0 = \frac{60,000}{2^5} = \frac{60,000}{32} = 1,875 \][/tex]
Thus, the initial population at [tex]\( t = 0 \)[/tex] is 1,875 bacteria.
### Step 2: Finding the Bacterial Population After 4 Hours
1. Convert the Time After 4 Hours to Minutes:
[tex]\[ 4 \text{ hours} = 4 \times 60 \text{ minutes} = 240 \text{ minutes} \][/tex]
2. Calculate the Total Time After 4 Hours from [tex]\( t = 100 \)[/tex]:
[tex]\[ \text{Total time} = 100 \text{ minutes} + 240 \text{ minutes} = 340 \text{ minutes} \][/tex]
3. Find the Number of Doublings from [tex]\( t = 100 \)[/tex] to [tex]\( t = 340 \)[/tex]:
[tex]\[ \text{Number of doublings} = \frac{240 \text{ minutes}}{20 \text{ minutes/doubling}} = 12 \][/tex]
4. Calculate the Population After 4 Hours:
Using the population at [tex]\( t = 100 \)[/tex] minutes as our starting point, and knowing it will double 12 times:
[tex]\[ \text{Population after 4 hours} = 60,000 \times (2^{12}) \][/tex]
5. Compute the Final Population:
[tex]\[ 60,000 \times 2^{12} = 60,000 \times 4,096 = 245,760,000 \][/tex]
Thus, the population after 4 hours is 245,760,000 bacteria.
### Summary:
- The initial population at [tex]\( t = 0 \)[/tex] is 1,875 bacteria.
- The bacterial population after 4 hours is 245,760,000 bacteria.
### Step 1: Finding the Initial Population at Time [tex]\( t = 0 \)[/tex]
1. Identify the Known Values:
- Doubling period: 20 minutes
- Given time (t): 100 minutes
- Bacterial population at [tex]\( t = 100 \)[/tex] minutes: 60,000
2. Calculate the Number of Doublings from [tex]\( t = 0 \)[/tex] to [tex]\( t = 100 \)[/tex]:
Since the doubling period is 20 minutes, the number of doublings that occur in 100 minutes is calculated as:
[tex]\[ \text{Number of doublings} = \frac{100 \text{ minutes}}{20 \text{ minutes/doubling}} = 5 \][/tex]
3. Determine the Initial Population Using the Doubling Formula:
The relationship between the initial population (P0) and the population at time [tex]\( t = 100 \)[/tex] minutes (P100) can be described by the formula:
[tex]\[ P_{100} = P_0 \times (2^{\text{Number of doublings}}) \][/tex]
Rearranging this formula to solve for [tex]\( P_0 \)[/tex]:
[tex]\[ P_0 = \frac{P_{100}}{2^{\text{Number of doublings}}} \][/tex]
4. Plug in the Known Values:
[tex]\[ P_0 = \frac{60,000}{2^5} = \frac{60,000}{32} = 1,875 \][/tex]
Thus, the initial population at [tex]\( t = 0 \)[/tex] is 1,875 bacteria.
### Step 2: Finding the Bacterial Population After 4 Hours
1. Convert the Time After 4 Hours to Minutes:
[tex]\[ 4 \text{ hours} = 4 \times 60 \text{ minutes} = 240 \text{ minutes} \][/tex]
2. Calculate the Total Time After 4 Hours from [tex]\( t = 100 \)[/tex]:
[tex]\[ \text{Total time} = 100 \text{ minutes} + 240 \text{ minutes} = 340 \text{ minutes} \][/tex]
3. Find the Number of Doublings from [tex]\( t = 100 \)[/tex] to [tex]\( t = 340 \)[/tex]:
[tex]\[ \text{Number of doublings} = \frac{240 \text{ minutes}}{20 \text{ minutes/doubling}} = 12 \][/tex]
4. Calculate the Population After 4 Hours:
Using the population at [tex]\( t = 100 \)[/tex] minutes as our starting point, and knowing it will double 12 times:
[tex]\[ \text{Population after 4 hours} = 60,000 \times (2^{12}) \][/tex]
5. Compute the Final Population:
[tex]\[ 60,000 \times 2^{12} = 60,000 \times 4,096 = 245,760,000 \][/tex]
Thus, the population after 4 hours is 245,760,000 bacteria.
### Summary:
- The initial population at [tex]\( t = 0 \)[/tex] is 1,875 bacteria.
- The bacterial population after 4 hours is 245,760,000 bacteria.
We appreciate your contributions to this forum. Don't forget to check back for the latest answers. Keep asking, answering, and sharing useful information. For dependable answers, trust IDNLearn.com. Thank you for visiting, and we look forward to assisting you again.
