Question

a petri dish starts off with 5 million bacteria. the number of bacteria grows at a continuous rate of 3.5% per minute. how many minutes will it take the petri dish to have 10 million bacteria? round your answer to the nearest minute.

Explanation:

Step1: Define continuous growth formula

The continuous growth formula is $N(t) = N_0 e^{rt}$, where $N(t)$ is the final population, $N_0$ is the initial population, $r$ is the growth rate, and $t$ is time.

Step2: Plug in given values

We know $N(t)=10$, $N_0=5$, $r=0.035$. Substitute into the formula:
$10 = 5e^{0.035t}$

Step3: Isolate the exponential term

Divide both sides by 5:
$\frac{10}{5} = e^{0.035t}$
$2 = e^{0.035t}$

Step4: Take natural log of both sides

$\ln(2) = \ln(e^{0.035t})$
Use $\ln(e^x)=x$ to simplify:
$\ln(2) = 0.035t$

Step5: Solve for t

Rearrange to solve for $t$:
$t = \frac{\ln(2)}{0.035}$
Calculate the value:
$t \approx \frac{0.6931}{0.035} \approx 19.80$

Step6: Round to nearest minute

Round 19.80 to the nearest whole number:
$t \approx 20$

Answer:

20 minutes