Problem 43 - Algebra and Trigonometry

Question

The logistic growth function $$ P(x)=\frac{90}{1+271 e^{-0.122 x}} $$ models the percentage, \(P(x),\) of Americans who are \(x\) years old with some coronary heart disease. Use the function to solve Exercises \(43-46\) What percentage of 20 -year-olds have some coronary heart disease?

Step-by-Step Solution

Answer

The percentage of 20-year-olds who have some coronary heart disease is approximately 3.67%

Step 1: Understanding the Problem

The given logistic function, \(P(x)=\frac{90}{1+271 e^{-0.122 x}}\), models the percentage of Americans who have coronary heart disease at age \(x\). The exercise asks to find the percentage of 20-year-olds with coronary heart disease.

Step 2: Substituting the age into the function

Substitute \(x = 20\) into \(P(x)=\frac{90}{1+271 e^{-0.122 x}}\), to find the percentage of 20-year olds who have coronary heart disease. This gives \(P(20) = \frac{90}{1+271 e^{-0.122*20}}\).

Step 3: Calculating the Percentage

Solve the equation from Step 2 to calculate the percentage. This gives \(P(20) = \frac{90}{1+271 e^{-2.44}}\). When evaluating the denominator, we have \( 1+271 e^{-2.44} = 1+271*0.0867 ≈ 24.5117 \). Thus, \(P(20) = \frac{90}{24.5117} ≈ 3.67 \)