Problem 107 - Algebra for College Students

Question

If a certain fabric is priced at \(\$ 3.98\) per yard, express the cost \(C(x)\) as a function of the number of yards \(x .\) Find \(C(3)\).

Step-by-Step Solution

Answer

The cost function is \(C(x) = 3.98x.\) For \(3\) yards, \(C(3) = 11.94.\)

Step 1 - Understanding the Problem

The price of fabric is given as \(3.98 per yard. We need to express the cost function \)C(x)\( representing the cost in terms of the number of yards, \)x.

Step 2 - Defining the Cost Function

The cost function, \(C(x)\), can be defined as the price per yard multiplied by the number of yards, \(x\). Thus, \(C(x) = 3.98 \times x.\)

Step 3 - Evaluating the Cost Function for \(x = 3\)

To find the cost for 3 yards, substitute \(x = 3\) into the cost function \(C(x)\). \(C(3) = 3.98 \times 3.\)

Step 4 - Performing the Calculation

Calculate \(C(3)\) by performing the multiplication: \(C(3) = 3.98 \times 3 = 11.94.\)

Key Concepts

linear equations

A linear equation forms the foundation of our cost function in this problem. Linear equations have the general form of \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. In our specific case, the cost of fabric per yard provides the slope, and since there is no fixed starting cost before buying any fabric, our y-intercept is \( 0 \).
So, our linear equation simplifies to \( C(x) = 3.98x \). Linear equations like these are straightforward to work with because they directly link two quantities through a constant rate, making them easy to graph and interpret.

function evaluation

Function evaluation involves calculating the value of a function for a given input. In the exercise, we are asked to find \( C(3) \), which means we need to evaluate the cost function when the number of yards (\( x \)) is \( 3 \).
Function evaluation follows these steps:

• Identify the given input (here, \( x = 3 \)).
• Substitute the input value into the function (\( C(x) = 3.98x \)).
• Perform the necessary arithmetic operations.

Let's substitute and compute:
\( C(3) = 3.98 \times 3 \).
This simple process of plugging in a value helps us understand how changes in one variable affect the output.

variable substitution

Variable substitution is integral for solving functions, as it allows us to insert specific values for the variables of interest. It involves:

• Recognizing the variable in the function (here, \( x \)).
• Inserting the desired value to see its effect (substituting \( x = 3 \)).

Applying this on our example, we substitute \( x \) with \( 3 \) in the cost function \( C(x) = 3.98x \), resulting in \( C(3) \). By performing this substitution, we directly calculate the cost for 3 yards of fabric as \( C(3) = 3.98 \times 3 \). This process is powerful because it transforms abstract functions into practical numerical values.

cost calculation

Cost calculation is the final step in this exercise, where we determine the total cost based on the number of units and the price per unit. Here, the unit price is \$3.98 per yard. By multiplying the number of units (yards) by the unit price, we get the total cost.
The formula is: \( C(x) = 3.98 \times x \). For \( x = 3 \), it becomes: \( C(3) = 3.98 \times 3 \). Performing this multiplication yields the total cost: \( C(3) = 11.94 \). Cost calculations like these are very common in everyday situations, helping us budget and make informed purchasing decisions.