Answer :
To solve the problem of finding the size of the bacterial population after 20 minutes and after 7 hours, we need to follow these steps:
1. Understanding the Growth Formula:
The bacteria culture grows according to the formula:
[tex]\[
p(t) = 3000 \, e^{kt}
\][/tex]
where [tex]\( p(t) \)[/tex] is the population at time [tex]\( t \)[/tex] in hours, and [tex]\( k \)[/tex] is a constant we need to determine.
2. Determining the Constant [tex]\( k \)[/tex]:
The problem states that the population doubles every half hour (0.5 hours). Therefore, after 0.5 hours, the population will be:
[tex]\[
2 \times 3000 = 6000
\][/tex]
Substitute into the equation:
[tex]\[
6000 = 3000 \, e^{k \times 0.5}
\][/tex]
Simplifying:
[tex]\[
2 = e^{0.5k}
\][/tex]
To solve for [tex]\( k \)[/tex], take the natural logarithm of both sides:
[tex]\[
\ln(2) = 0.5k
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{\ln(2)}{0.5} \approx 1.3862943611198906
\][/tex]
3. Finding the Population After 20 Minutes:
Convert 20 minutes into hours:
[tex]\[
\text{Time in hours} = \frac{20}{60} = \frac{1}{3} \text{ hours}
\][/tex]
Substitute into the population formula:
[tex]\[
p\left(\frac{1}{3}\right) = 3000 \, e^{1.3862943611198906 \times \frac{1}{3}}
\][/tex]
This results in a population size of approximately 4762 bacteria.
4. Finding the Population After 7 Hours:
Substitute 7 hours into the population formula:
[tex]\[
p(7) = 3000 \, e^{1.3862943611198906 \times 7}
\][/tex]
This results in a population size of approximately 49,152,000 bacteria.
Therefore, the size of the bacterial population after 20 minutes is approximately 4762, and after 7 hours, it is approximately 49,152,000.
1. Understanding the Growth Formula:
The bacteria culture grows according to the formula:
[tex]\[
p(t) = 3000 \, e^{kt}
\][/tex]
where [tex]\( p(t) \)[/tex] is the population at time [tex]\( t \)[/tex] in hours, and [tex]\( k \)[/tex] is a constant we need to determine.
2. Determining the Constant [tex]\( k \)[/tex]:
The problem states that the population doubles every half hour (0.5 hours). Therefore, after 0.5 hours, the population will be:
[tex]\[
2 \times 3000 = 6000
\][/tex]
Substitute into the equation:
[tex]\[
6000 = 3000 \, e^{k \times 0.5}
\][/tex]
Simplifying:
[tex]\[
2 = e^{0.5k}
\][/tex]
To solve for [tex]\( k \)[/tex], take the natural logarithm of both sides:
[tex]\[
\ln(2) = 0.5k
\][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[
k = \frac{\ln(2)}{0.5} \approx 1.3862943611198906
\][/tex]
3. Finding the Population After 20 Minutes:
Convert 20 minutes into hours:
[tex]\[
\text{Time in hours} = \frac{20}{60} = \frac{1}{3} \text{ hours}
\][/tex]
Substitute into the population formula:
[tex]\[
p\left(\frac{1}{3}\right) = 3000 \, e^{1.3862943611198906 \times \frac{1}{3}}
\][/tex]
This results in a population size of approximately 4762 bacteria.
4. Finding the Population After 7 Hours:
Substitute 7 hours into the population formula:
[tex]\[
p(7) = 3000 \, e^{1.3862943611198906 \times 7}
\][/tex]
This results in a population size of approximately 49,152,000 bacteria.
Therefore, the size of the bacterial population after 20 minutes is approximately 4762, and after 7 hours, it is approximately 49,152,000.
