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 percentage of 35 - year - olds who have some coronary heart disease. what is the percentage of 35 - year - olds with some coronary heart disease? (round to one decimal place.)

Explanation:

Step1: Substitute x = 35 into the function

We have the function \( P(x)=\frac{90}{1 + 271e^{-0.122x}} \). Substitute \( x = 35 \) into it:
\( P(35)=\frac{90}{1 + 271e^{-0.122\times35}} \)

Step2: Calculate the exponent part

First, calculate \( -0.122\times35=-4.27 \)

Step3: Calculate the exponential term

Calculate \( e^{-4.27} \). Using a calculator, \( e^{-4.27}\approx0.0149 \)

Step4: Calculate the denominator

Calculate \( 1 + 271\times0.0149 \). First, \( 271\times0.0149 = 271\times0.01 + 271\times0.0049 = 2.71+1.3279 = 4.0379 \). Then, \( 1 + 4.0379 = 5.0379 \)

Step5: Calculate the final value

Now, \( P(35)=\frac{90}{5.0379}\approx17.9 \)

Answer:

17.9