Question

the logistic growth function $p(x)=\frac{90}{1 + 271e^{-0.122x}}$ models the percentage, $p(x)$, of americans who are x years old and have some coronary heart disease. use this function to find the age at which the percentage of americans who have some coronary heart disease is 10%. what is the age at which 10% of americans have some coronary heart disease? years old (round to the nearest integer.)

Explanation:

Step1: Set up the equation

We are given $P(x) = 10$ and $P(x)=\frac{90}{1 + 271e^{-0.122x}}$, so $10=\frac{90}{1 + 271e^{-0.122x}}$.

Step2: Cross - multiply

$10(1 + 271e^{-0.122x})=90$. Then $1 + 271e^{-0.122x}=9$.

Step3: Isolate the exponential term

$271e^{-0.122x}=9 - 1=8$. So $e^{-0.122x}=\frac{8}{271}$.

Step4: Take the natural logarithm of both sides

$\ln(e^{-0.122x})=\ln(\frac{8}{271})$. Using the property $\ln(e^{a}) = a$, we get $-0.122x=\ln(\frac{8}{271})$.

Step5: Solve for x

$x=\frac{\ln(\frac{8}{271})}{- 0.122}$. Calculate $\ln(\frac{8}{271})\approx\ln(8)-\ln(271)\approx2.0794 - 5.6009=-3.5215$. Then $x=\frac{-3.5215}{-0.122}\approx29$.

Answer:

29