Problem 44 - College Algebra

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 80 -year-olds have some coronary heart disease?

Step-by-Step Solution

Answer

Approximately 88.18% of 80 year olds have some coronary heart disease.

Step 1: Identify the given values

The given logistic function is \(P(x)=\frac{90}{1+271 e^{-0.122 x}}\) and the age asked for is \(x = 80\).

Step 2: Substitute value of x

Replace \(x\) in the logistic function \(P(x)\) by 80 to get a new equation \(P(80)=\frac{90}{1+271 e^{-0.122 \cdot 80}}\).

Step 3: Solve the equation

Calculate the expression to find the value of \(P(80)\). Using a calculator, \(P(80)\) evaluates approximately to 88.1831.

Key Concepts

Coronary Heart Disease

Coronary Heart Disease (CHD), also known as coronary artery disease, is a common type of heart disease. It occurs when the coronary arteries, which supply blood to the heart muscle, become narrowed or blocked. This is often due to a build-up of plaque, a substance made of cholesterol, fatty deposits, and other materials. When the heart doesn't get enough blood, it can't receive enough oxygen, which can lead to chest pain, shortness of breath, or heart attacks.

Individuals of various age groups can be affected by CHD, but the risk typically increases with age. This exercise explores the likelihood of developing CHD through a logistic growth model, which helps predict how many people in different age groups may have some form of this disease. The focus here is understanding the percentage of 80-year-olds impacted by CHD, providing valuable insight into how age influences heart health.

Percentage Calculation

When solving real-world problems such as determining the percentage of people at a certain age with coronary heart disease, percentage calculations are crucial. The logistic growth function provided, \[ P(x)=\frac{90}{1+271 e^{-0.122 x}} \]helps model these percentages effectively. To find the specific percentage of a certain age group, we simply need to substitute the age into the function and solve for that particular value.

In this instance, the formula gives us the predicted percentage of 80-year-olds who have some form of coronary heart disease. Percentage calculations are essential in many fields, such as statistics, finance, and health sciences, providing a clear, numeric representation of data as a part of a whole.

Substitution in Functions

Substitution in functions is a method where you replace a variable with a given value. This technique is used to evaluate functions for specific instances, such as determining exact values.

In the problem presented, we want to know the predicted percentage of 80-year-olds with coronary heart disease. Here, the variable is the age, represented by \( x \) in the equation \( P(x)=\frac{90}{1+271 e^{-0.122 x}} \). By substituting \( x = 80 \), we transform the general function into a specific equation, \( P(80)=\frac{90}{1+271 e^{-0.122 \cdot 80}} \).

This process allows us to calculate the result numerically, thus providing an exact percentage for that age group. Substitution is a foundational concept in algebra that helps bridge symbolic mathematics with real-world applications.