Question

a bacteria culture containing 430 bacteria is started in a petri dish. after 4 hours, the bacteria population has grown to 1075. assume the bacteria growth is exponential. what is the growth rate of the bacteria? round to 2 decimals. if a(t) is the population of bacteria at hour t, create a model for the population of bacteria.

Explanation:

Step1: Recall the exponential - growth formula

The general formula for exponential growth of a population is $A(t)=A_0e^{rt}$, where $A_0$ is the initial population, $r$ is the growth rate, and $t$ is the time. We know that $A_0 = 430$, $A(4)=1075$, and $t = 4$.

Step2: Substitute the values into the formula

Substitute $A_0 = 430$, $A(4)=1075$, and $t = 4$ into $A(t)=A_0e^{rt}$, we get $1075 = 430e^{4r}$.

Step3: Solve for $r$

First, divide both sides of the equation by 430: $\frac{1075}{430}=e^{4r}$. Since $\frac{1075}{430}=2.5$, the equation becomes $2.5 = e^{4r}$.
Then, take the natural - logarithm of both sides: $\ln(2.5)=\ln(e^{4r})$.
Using the property $\ln(e^x)=x$, we have $\ln(2.5)=4r$.
Now, solve for $r$: $r=\frac{\ln(2.5)}{4}$.
We know that $\ln(2.5)\approx0.916291$, so $r=\frac{0.916291}{4}\approx0.23$.

Step4: Create the population model

The population model is $A(t)=430e^{0.23t}$.

Answer:

The growth rate $r\approx0.23$ and the population model is $A(t)=430e^{0.23t}$