Question

a scientist writes the equation n(h) = 100e^{0.25h} to model the growth of a certain bacteria in a petri dish, where n represents the number of bacteria after h hours. after approximately how many hours will 450 bacteria be present? round your answer to the nearest whole number.
1 hour
6 hours
13 hours
15 hours

Explanation:

Step1: Substitute $N(h)$ value

We are given $N(h)=100e^{0.25h}$ and $N(h) = 450$. So, $450=100e^{0.25h}$.

Step2: Isolate the exponential term

Divide both sides by 100: $\frac{450}{100}=e^{0.25h}$, which simplifies to $4.5 = e^{0.25h}$.

Step3: Take natural - logarithm of both sides

$\ln(4.5)=\ln(e^{0.25h})$. Since $\ln(e^{x}) = x$, we have $\ln(4.5)=0.25h$.

Step4: Solve for $h$

$h=\frac{\ln(4.5)}{0.25}$. We know that $\ln(4.5)\approx1.5041$. Then $h=\frac{1.5041}{0.25}=6.0164\approx6$.

Answer:

6 hours