We are tasked with finding the area of a rectangular dog run that Elissa is creating with 28 feet of fencing, where the length of the run is denoted as $x$ feet. The goal is to express the area as a function of $x$.

Step 1: Understanding the Problem

The perimeter of a rectangular shape is given by: $\text{Perimeter} = 2 \times (\text{length} + \text{width})$ In this case, the length is $x$, and let's call the width $w$.

Given the total fencing available is 28 feet, the equation for the perimeter becomes: $2(x + w) = 28$ Solving for $w$, we get: $x + w = 14 \quad \Rightarrow \quad w = 14 - x$

Step 2: Expression for Area

The area $A(x)$ of a rectangle is given by: $A(x) = \text{length} \times \text{width} = x \times (14 - x)$ Expanding this expression: $A(x) = 14x - x^2$

Step 3: Verifying the Options

The correct expression for the area as a function of $x$ is: $A(x) = 14x - x^2$ Thus, the correct answer is: $\boxed{A(x) = 14x - x^2}$

Do you want more details or have any questions?

Here are 5 related questions you might explore:

1. What are the dimensions of the dog run if the length is 10 feet?
2. How would the area formula change if Elissa had more fencing?
3. Can you find the value of $x$ that maximizes the area of the dog run?
4. What is the maximum possible area of the dog run using the given fencing?
5. How would you model this problem if the shape of the dog run were circular instead of rectangular?

Tip: When working with optimization problems involving fencing, always check if the perimeter constraint is being correctly applied to derive the width or other dimensions.