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.
