Question

a culture of bacteria grows according to the continuous growth model b = f(t)=700e^{0.069t} where b is the number of bacteria and t is in hours. find f(0)=______ to the nearest whole number, find the number of bacteria after 8 hours. to the nearest tenth of an hour, determine how long take the population to grow to 900 bacteria.

Explanation:

Step1: Find $f(0)$

Substitute $t = 0$ into $f(t)=700e^{0.069t}$. Since $e^{0}=1$, then $f(0)=700\times e^{0}=700\times1 = 700$.

Step2: Find the number of bacteria after 8 hours

Substitute $t = 8$ into $f(t)=700e^{0.069t}$. So $f(8)=700e^{0.069\times8}=700e^{0.552}$. Using a calculator, $e^{0.552}\approx1.736$, and $700\times1.736 = 1215.2\approx1215$.

Step3: Find the time when the population is 900

Set $f(t)=900$, so $900 = 700e^{0.069t}$. First, divide both sides by 700: $\frac{900}{700}=e^{0.069t}$, which simplifies to $\frac{9}{7}=e^{0.069t}$. Then take the natural - logarithm of both sides: $\ln(\frac{9}{7})=\ln(e^{0.069t})$. Since $\ln(e^{x})=x$, we have $\ln(\frac{9}{7}) = 0.069t$. Calculate $\ln(\frac{9}{7})\approx\ln(1.286)\approx0.255$. Then $t=\frac{\ln(\frac{9}{7})}{0.069}=\frac{0.255}{0.069}\approx3.7$.

Answer:

$f(0)=700$
The number of bacteria after 8 hours is 1215
It takes approximately 3.7 hours for the population to grow to 900 bacteria.