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 40%.

what is the age at which 40% of americans have some coronary heart disease?
\boxed{} years old (round to the nearest integer.)

Explanation:

Step1: Set up the equation

We know that \( P(x) = 40 \), so we set up the equation \( \frac{90}{1 + 271e^{-0.122x}} = 40 \).

Step2: Solve for the denominator

Multiply both sides by \( 1 + 271e^{-0.122x} \) to get \( 90 = 40(1 + 271e^{-0.122x}) \). Then divide both sides by 40: \( \frac{90}{40}=1 + 271e^{-0.122x} \), which simplifies to \( 2.25 = 1 + 271e^{-0.122x} \).

Step3: Isolate the exponential term

Subtract 1 from both sides: \( 2.25 - 1=271e^{-0.122x} \), so \( 1.25 = 271e^{-0.122x} \). Then divide both sides by 271: \( e^{-0.122x}=\frac{1.25}{271}\approx0.0046125 \).

Step4: Take the natural logarithm

Take the natural logarithm of both sides: \( \ln(e^{-0.122x})=\ln(0.0046125) \). Using the property \( \ln(e^a)=a \), we get \( - 0.122x=\ln(0.0046125) \).

Step5: Solve for x

Divide both sides by - 0.122: \( x=\frac{\ln(0.0046125)}{- 0.122}\approx\frac{-5.37}{-0.122}\approx44 \).

Answer:

44