Problem 43 - 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 20 -year-olds have some coronary heart disease?

Step-by-Step Solution

Answer

The percentage of 20 -year-olds with some coronary heart disease is approximately 6.11%.

Step 1: Identify the Required Age

The required age is directly given in the question as 20 years. So, \(x = 20\).

Step 2: Substitute the Value of x into the Function

Substitute \(x = 20\) into the logistic growth function \(P(x)=\frac{90}{1+271 e^{-0.122 x}}\). After this substitution, it becomes \(P(20)=\frac{90}{1+271 e^{-0.122 \cdot 20}}\).

Step 3: Calculate the Value of the Function

Next, calculate the value of the function. Parentheses are needed around the -0.122 times 20, because multiplication should be performed before the exponent is computed. Note that \(e^{-0.122 \cdot 20}\) is approximated to eight decimal places to ensure enough precision.

Step 4: Final Computation

Finally perform the calculation within the function. The expression in the denominator of the function should be computed first, before dividing the result into 90 to give the final answer.

Key Concepts

Exponential Function

The logistic growth model employs exponential functions to describe scenarios where growth accelerates rapidly at first and then slows as it approaches a maximum carrying capacity. In the context of coronary heart disease, this function helps estimate the percentage of individuals affected as they age. An exponential function is typically expressed as:

• The base (in this case, the constant `e`, or the mathematical constant approximately equal to 2.71828).
• The exponent, which dictates the rate of growth or decay (in this case, `-0.122 x`). negative exponents represent decay, meaning that as the age `x` increases, certain characteristics or variables might decrease at a set rate.

In our logistic model, this decay rate helps determine how the baseline percentage of coronary heart disease changes as age progresses. This is critical when dealing with real-world data affected by numerous variables.

Percentage Calculation

Percentage calculations are useful in comparing the relative sizes of different values, often providing insights into proportions of populations affected by specific conditions. In the exercise given, the percentage of people affected by coronary heart disease at a particular age is determined by the logistic function.

• Calculations involve substituting a specific age (in this case, 20) into the equation.
• This requires solving the function to find out what fraction of 100 represents individuals with heart disease.

Once all substitutions and calculations are completed, the output is converted into a percentage, offering a straightforward numeric representation of how widespread the condition is among individuals of that specific age.

Coronary Heart Disease

Coronary heart disease (CHD) is a condition marked by damage or disease in the heart's major blood vessels. Typically influenced by factors such as diet, lifestyle, and genetics, CHD can lead to serious complications like heart attacks or heart failure.

• As people age, the risk of developing coronary heart disease generally increases.
• The logistic function in the exercise provides a way to predict and analyze how this risk escalates with age, providing valuable data for public health assessments.

By using mathematical modeling, researchers and healthcare professionals can better comprehend and mitigate risks associated with CHD, focusing efforts on preventative care across age groups.

Age Variable

The age variable in our logistic growth model plays a crucial role in determining the susceptibility to coronary heart disease within a population.

• In the logistic function, the 'age' is represented as `x`, the independent variable that influences the outcome.
• As `x` increases, reflecting an increasing age, the function modulates the percentage of individuals affected by coronary heart disease.

This variable emphasizes how aging impacts the health conditions being studied. By understanding the role of age in our model, we gain insights into larger patterns of disease prevalence as people get older. This knowledge can inform policies and interventions aimed at reducing age-related health risks.