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 the percentage of 67 - year olds who have some coronary heart disease. what is the percentage of 67 - year olds with some coronary heart disease?
$\square\\%$ (round to one decimal place.)

Explanation:

Step1: Substitute \(x = 67\) into the function

We have the logistic - growth function \(P(x)=\frac{90}{1 + 271e^{-0.122x}}\). Substitute \(x = 67\) into the function:

$$P(67)=\frac{90}{1+271e^{-0.122\times67}}$$

First, calculate the exponent: \(-0.122\times67=-8.174\)
Then, calculate \(e^{-8.174}\). Using the formula \(y = e^{x}\), where \(x=-8.174\), we know that \(e^{-8.174}=\frac{1}{e^{8.174}}\approx\frac{1}{3550.79}\approx0.000282\)

Step2: Calculate the denominator

Calculate \(1 + 271e^{-0.122\times67}\). Substitute \(e^{-0.122\times67}\approx0.000282\) into the denominator:
\(1+271\times0.000282=1 + 0.076422=1.076422\)

Step3: Calculate \(P(67)\)

Now, calculate \(P(67)=\frac{90}{1.076422}\approx83.6\)

Answer:

\(83.6\%\)