To predict the number of bacteria after 17 hours, we'll use the exponential growth formula:
[tex]\[ P = A e^{k t} \][/tex]
where:
- [tex]\( P \)[/tex] is the population at time [tex]\( t \)[/tex],
- [tex]\( A \)[/tex] is the initial population,
- [tex]\( k \)[/tex] is the growth rate constant,
- [tex]\( t \)[/tex] is the time.
1. Initial data:
- Initial bacteria count ([tex]\( A \)[/tex]): 2000
- Bacteria count after 4 hours: 2600
- Time ([tex]\( t \)[/tex]): 4 hours
2. Determine the growth rate constant ([tex]\( k \)[/tex]):
We know that after 4 hours, the bacteria count is 2600. Using the equation:
[tex]\[ 2600 = 2000 e^{4k} \][/tex]
3. Solve for [tex]\( k \)[/tex]:
Divide both sides by 2000:
[tex]\[ \frac{2600}{2000} = e^{4k} \][/tex]
Simplifying:
[tex]\[ 1.3 = e^{4k} \][/tex]
Take the natural logarithm of both sides:
[tex]\[ \ln(1.3) = 4k \][/tex]
Solve for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{\ln(1.3)}{4} \][/tex]
4. Predict the population after 17 hours:
Now we need to find the bacteria count after 17 hours. Using the formula:
[tex]\[ P = 2000 e^{k \cdot 17} \][/tex]
Substitute [tex]\( k \)[/tex] with the value obtained:
[tex]\[ k = \frac{\ln(1.3)}{4} \][/tex]
Therefore:
[tex]\[ P = 2000 e^{\left(\frac{\ln(1.3)}{4}\right) \cdot 17} \][/tex]
5. Calculate the result:
The resulting value for [tex]\( P \)[/tex] after rounding to the nearest whole number is:
[tex]\[ P \approx 6099 \][/tex]
So, after 17 hours, the predicted number of bacteria will be approximately 6,099.
