Problem 44 - College Algebra Essentials

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 44.79% of 80-year-olds have some coronary heart disease.

Step 1: Understand the Function

The given function \(P(x)=\frac{90}{1+271 e^{-0.122 x}}\) models the percentage of Americans with coronary heart disease in relation to their age. Here, \(x\) represents the age.

Step 2: Substitute the Given Age

Substitute \(x = 80\) into the function to find out the percentage of 80-year-olds with some coronary heart disease. Hence, \(P(80) =\frac{90}{1+271 e^{-0.122 * 80}}\).

Step 3: Simplify the Expression to Get the Answer

Solve the equation to get the percentage. Using a calculator or mathematical software, the result should be approximately 44.79%.

Key Concepts

Percentage Calculation

The logistic growth function in this exercise offers a practical use of percentage calculation. The purpose here is to compute a percentage based on a given formula that models real-world data.

In the formula, the output, or result, of the function is in percentage form. To calculate this percentage, the function requires substitution. Every age, denoted by \( x \), aligns with a specific percentage of people dealing with coronary heart disease. The task is simply substituting \( x = 80 \) into the equation and calculating \( P(80) \).

When you perform the computation, remember to tackle it step-by-step:

• First, substitute \( x = 80 \) to have \, \( P(80) = \frac{90}{1 + 271 e^{-0.122 \times 80}} \).
• Next, use a calculator to exponentiate and simplify \( e^{-0.122 \times 80} \).
• After simplifying, perform the division to arrive at the final percentage, 44.79%.

Computing this percentage relies on understanding the sequence of operations, ensuring accurate substitution and calculation to derive meaningful data.

Coronary Heart Disease

Coronary heart disease (CHD) is a significant health concern. It involves problems with the arteries and heart muscle due to plaque buildup. As people age, the risk often increases, which is relevant to this logistic growth model.

This model helps researchers and policymakers understand how CHD prevalence shifts with age. The growth function shows the trend seen in the population percentage suffering from heart disease as age changes. By understanding this distribution, we can aim for better healthcare measures in susceptible age groups.

Data-driven approaches, like the one used in this exercise, allow us to:

• Identify high-risk groups and focus preventive measures.
• Address lifestyle factors and medical interventions for at-risk populations.
• Design public health strategies that reduce incidence rates.

Interpreting the model offers insights into connections between age and coronary heart disease, facilitating an informed perspective on health risks as one ages.

Exponential Functions

Exponential functions are mathematical expressions where variables appear in the exponent. They can model growth and decay in many real-world scenarios, from population dynamics to radioactive decay.

In this logistic growth function, the term \( e^{-0.122 \times x} \) represents the proportion of the population without the disease as age \( x \) increases. When \( x \) is small, \( e^{-0.122 \times x} \) is large, implying a small denominator, which results in a smaller percentage of coronary heart disease. As \( x \) increases, \( e^{-0.122 \times x} \) decreases, enlarging the denominator and reflecting the higher incidences of the disease.

The logistic growth model relies on:

• Balancing the initial exponential component observed in simpler growth models.
• Adjusting real-life constraints and saturation levels as provided by the logistic framework.
• Providing a realistic boundary to the predicted output—in this case, a percentage less than or equal to 100%.

This function encapsulates the idea that growth slows as capacity limits are approached, characteristic of many natural processes and important for understanding phenomena like disease spread.