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 60-year olds who have some coronary heart disease.

what is the percentage of 60-year olds with some coronary heart disease?
\boxed{}% (round to one decimal place.)

Explanation:

Step1: Substitute x = 60 into the function

We have the logistic growth function \( P(x)=\frac{90}{1 + 271e^{-0.122x}} \). Substitute \( x = 60 \) into the function:
\( P(60)=\frac{90}{1+271e^{-0.122\times60}} \)

Step2: Calculate the exponent

First, calculate the exponent \( - 0.122\times60=-7.32 \)
So the function becomes \( P(60)=\frac{90}{1 + 271e^{-7.32}} \)

Step3: Calculate \( e^{-7.32} \)

We know that \( e^{-7.32}\approx0.00066 \) (using a calculator to find the value of the exponential function)

Step4: Calculate the denominator

Calculate \( 271\times0.00066\approx0.1789 \)
Then the denominator \( 1 + 0.1789 = 1.1789 \)

Step5: Calculate the value of P(60)

Now, \( P(60)=\frac{90}{1.1789}\approx76.4 \) (rounded to one decimal place)

Answer:

\( 76.4 \)