Problem 43 - Introductory Algebra for College Students

Question

The function \(f(x)=0.76 x+171.4\) models the cholesterol level of an American man as a function of his age, \(x,\) in years. Find and interpret \(f(20)\)

Step-by-Step Solution

Answer

The cholesterol level of an American man who is 20 years old is expected to be 186.6 milligrams per deciliter.

Step 1: Plug In The Value

Substitute \(x = 20\) into the function \(f(x)=0.76 x+171.4\) to get \(f(20)=0.76 \cdot 20+171.4\)

Step 2: Compute The Expression

Now perform the arithmetic to get the value for \(f(20)\), \(f(20)=0.76 \cdot 20+171.4 = 15.2 + 171.4 = 186.6\)

Step 3: Interpret The Result

The computed value represents the cholesterol level. Therefore, an American man who is 20 years old is expected to have a cholesterol level of 186.6 milligrams per deciliter.

Key Concepts

Function Evaluation

When we talk about function evaluation, we refer to the process of determining the output of a function based on a specific input. In the context of algebra, a function is a relationship that assigns exactly one output for each input. In our exercise, the linear function is expressed as f(x) = 0.76x + 171.4. To evaluate this function for x = 20, we simply replace every occurrence of x with 20 in the function's equation.

Here’s how this looks step by step:

1. Substitute the input value: We input x = 20 into the equation, yielding f(20) = 0.76(20) + 171.4.
2. Perform the arithmetic: Next, we multiply 0.76 by 20 to get 15.2, and then we add this to 171.4. The equation now reads f(20) = 186.6.

This process is powerful as it allows us to predict outcomes based on the function's model. Function evaluation can be applied not only in algebra but in various real-world scenarios, such as predicting population growth, calculating interest rates, or, as in our exercise, estimating cholesterol levels.

Interpreting Linear Models

Interpreting linear models is a fundamental part of understanding how algebra applies to the real world. A linear model is an equation that represents a straight line on a graph and is used to predict or explain a relationship between two variables. The general form of a linear model is y = mx + b, where y represents the dependent variable, x is the independent variable, m is the slope of the line (rate of change), and b is the y-intercept (the starting value when x is zero).

Here's what interpreting our linear model looks like:

1. Identify the components: In our exercise, the slope m is 0.76, which indicates that for each year increase in age, the cholesterol level rises by about 0.76 milligrams per deciliter.
2. Understand the constant: The y-intercept b is 171.4, suggesting the baseline cholesterol level independent of age.
3. Make predictions: Using this model, you can predict the cholesterol level for an American man at different ages (x values).
4. Analyze the relevance: When we calculate f(20) = 186.6, it means that a 20-year-old man is expected to have a cholesterol level of 186.6 milligrams per deciliter according to this model.

The interpretation of linear models helps us make sense of data and can even inform health guidelines or policies based on predicted trends over time.

Algebraic Expressions

In algebra, an algebraic expression is a combination of numbers, variables, and operation symbols that represents a mathematical relationship. Expressions can vary in complexity and do not contain equality or inequality signs, unlike equations. The elements of an algebraic expression include constants (fixed values), variables (placeholders for numbers), coefficients (numbers multiplied by variables), and an assortment of operations (addition, subtraction, multiplication, division).

For instance, in the expression f(x) = 0.76x + 171.4, 171.4 is a constant (the starting cholesterol level), x is a variable (the person's age), and 0.76 is the coefficient indicating the rate at which cholesterol level changes per year. To effectively work with algebraic expressions, one must:

1. Interpret the terms and their relationships.
2. Perform operations according to the correct order of operations (PEMDAS/BODMAS).
3. Substitute values correctly when evaluating the function.

Algebraic expressions are the building blocks for creating models that help us explore and understand relationships within various fields, including science, finance, and social sciences.