Answer:
About 24 hours.
Step-by-step explanation:
We are given that the experiment initially had 100 bacteria cells, and that its populations double every 15 hours.
From the given formula:
[tex]\displaystyle P_t=P_0(2)^\dfrac{t}{d}[/tex]
We can substitute 100 for P₀ and 15 for t. This yields:
[tex]\displasytyle P_t=100(2)^\dfrac{t}{15}[/tex]
We want to find after how many hours t will the population reach 300.
So, let's substitute 300 for P_t and solve for t:
[tex]\displaystyle 300=100(2)^\dfrac{t}{15}[/tex]
Divide:
[tex]3=(2)^\dfrac{t}{15}[/tex]
We can take the natural log of both sides:
[tex]\ln(3)=\ln\left(2^\dfrac{t}{15}\right)[/tex]
Logarithm properties:
[tex]\displaystyle \ln(3)=\frac{t}{15}\ln(2)[/tex]
Solve for t:
[tex]\displaystyle t=\frac{15\ln(3)}{\ln(2)}[/tex]
Use a calculator:
[tex]t\approx23.77[/tex]
In conclusion, the population will reach 300 bacteria after about 24 hours.
