Problem 62 - Introductory and Intermediate Algebra for College Students 4th

Question

The function$$f(x)=\frac{90}{1+270 e^{-0.122 x}}$$ models the percentage, \(f(x)\), of people \(x\) years old with some coronary heart disease. Use this function to solve exercise. Round answers to the nearest tenth of a percent. Evaluate \(f(70)\) and describe what this means in practical terms.

Step-by-Step Solution

Answer

To obtain the final result replace \(x\) with \(70\) in the equation, calculate the expression, and interpret the result. The result is the percentage of people, aged 70, that have some form of coronary heart disease.

Step 1: Understand the Function

The function \(f(x) = \frac{90}{1+270e^{-0.122x}}\) models the percentage of people of age \(x\) with coronary heart disease.

Step 2: Substitution

Now, substitute \(x = 70\) into the function. This gives \(f(70) = \frac{90}{1+270e^{-0.122(70)}}\)

Step 3: Calculation

Next, calculate the exponent part. This gives \(f(70) = \frac{90}{1+270e^{-8.54}}\). Further calculate to achieve the result.

Step 4: Interpretation

Finally, the result achieved represents the percentage of people of age 70 that suffer from coronary heart disease.

Key Concepts

Exponential Decay

Exponential decay is a process where quantities decrease at a rate proportional to their current value. This process is ubiquitous in nature and it is commonly described by exponential functions, which are characterized by their base, often represented as 'e' in the natural exponential function. This principle plays a pivotal role in health science, particularly when examining the spread of diseases, decay of biological specimens, or even the effectiveness of drugs over time.
The algebraic function given in the exercise, \(f(x) = \frac{90}{1+270e^{-0.122x}}\), portrays a type of exponential decay. Here, the exponent, \( -0.122x \), suggests a decline in the percentage of people with coronary heart disease as the age \(x\) increases. This decay reflects the medical reality that certain health risks can reduce significantly as a specific age group advances. In health sciences, understanding exponential decay allows for modeling complex biological processes and potential prognostic evaluations.

Mathematical Modeling

Mathematical modeling is a powerful tool for simplifying and analyzing real-world phenomena using the language of mathematics. In health sciences, mathematical models are vital for predicting disease trends, calculating medication dosages, and simulating biological processes. The algebraic function from the exercise, \(f(x)\), is a fine example of a mathematical model as it encapsulates the relationship between age \(x\) and the prevalence of a health condition—in this case, coronary heart disease.
Such models are formulated by identifying patterns or correlations in observational data and then constructing a function that describes these trends. The function \(f(x)\) appears to have been derived from epidemiological data and is being used to predict the percentage of individuals affected by heart disease at various ages. The use of exponential functions in modeling is particularly common since many biological systems exhibit exponential growth or decay.

Function Evaluation

Function evaluation is the process by which we calculate the output of a function for a given input. It is an essential exercise in algebra that lays the groundwork for more complex mathematical analysis. In the context of health science, evaluating a function may be used to determine specific health metrics based on patient data or demographic variables.
In executing the step-by-step solution provided, evaluating \(f(70)\) involves inputting the age '70' into the function to calculate the percentage of people with coronary heart disease. This evaluation is not purely theoretical; it has practical implications. Once calculated, the value \(f(70)\), expressed as a percentage, tells health professionals the extent of the disease's impact on a community's elderly population, allowing for targeted health interventions, resource allocation, and policy-making. Simplifying expressions and understanding the practical interpretation of the results are fundamental skills in applying algebraic functions in health and medical fields.