Question

the population of turtles at a nature preserve is modelled by the function p(t)=\frac{285}{1 + 8e^{-0.024t}} turtles, where t measures the time, in weeks, since cleanup of a nearby hazardous waste dump began. what is the t - coordinate of the inflection point of this model? round your answer to two decimal places.

Explanation:

Step1: Recall inflection - point formula for logistic function

For a logistic function $P(t)=\frac{L}{1 + Ae^{-kt}}$, the inflection - point occurs at $t=\frac{\ln(A)}{k}$. Here, $L = 285$, $A = 8$, and $k=0.024$.

Step2: Calculate the $t$ - coordinate of the inflection point

Substitute $A = 8$ and $k = 0.024$ into the formula $t=\frac{\ln(A)}{k}$. We know that $\ln(8)=\ln(2^{3}) = 3\ln(2)\approx3\times0.6931 = 2.0793$. Then $t=\frac{\ln(8)}{0.024}=\frac{2.0793}{0.024}$.

Step3: Perform the division

$t=\frac{2.0793}{0.024}=86.64$.

Answer:

$86.64$