- Identify the fixed cost of \(\$5\) as the y-intercept (\(b = 5\)) and the variable cost of \(\$2.25\) per pound as the slope (\(m = 2.25\)).
- Apply the linear slope-intercept form \(f(x) = mx + b\).
- Substitute the values into the equation to get the final function: \(f(x) = 2.25x + 5\).
- The correct equation is \(\boxed{f(x) = 2.25x + 5}\).
Explanation
- Problem Analysis
To determine the correct shipping cost function, we first analyze the components of the cost. The problem states that the cost of shipping a box is comprised of a fixed packing fee of \(\$5\) and a variable cost of \(\$2.25\) per pound. We are told to let \(x\) represent the weight of the box in pounds.
- Identifying Linear Components
A linear function is generally represented by the slope-intercept form: \(f(x) = mx + b\). In this context, \(m\) represents the rate of change or the variable cost per unit (per pound), and \(b\) represents the initial or fixed cost (the y-intercept). Given that the variable cost is \(\$2.25\) per pound, we have the slope \(m = 2.25\). Given that the fixed packing fee is \(\$5\), we have the y-intercept \(b = 5\).
- Constructing the Equation
By substituting the slope \(m = 2.25\) and the y-intercept \(b = 5\) into the slope-intercept form, we get:
\[f(x) = 2.25x + 5\]
This equation shows that for any weight \(x\), the total cost is the sum of the product of the rate and the weight, plus the fixed fee.
- Conclusion
Comparing our constructed equation to the given options, we see that \(f(x) = 2.25x + 5\) corresponds to the first option provided. Therefore, the function that represents the total shipping cost is \(\boxed{f(x) = 2.25x + 5}\).
Examples
This concept of linear relationships is essential in everyday life, particularly in budgeting and commerce. For example, when you take a taxi, there is often a starting fee (fixed cost) plus a rate per mile (variable cost). If the starting fee is \(\$5\) and the charge per mile is \(\$2.25\), the cost of a trip of \(x\) miles would be represented by the same linear function \(f(x) = 2.25x + 5\). Understanding this allows you to estimate your expenses accurately in various real-world situations.