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].
