Math Problem Statement
Solve
Solution
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 feet. The goal is to express the area as a function of .
Step 1: Understanding the Problem
The perimeter of a rectangular shape is given by: In this case, the length is , and let's call the width .
Given the total fencing available is 28 feet, the equation for the perimeter becomes: Solving for , we get:
Step 2: Expression for Area
The area of a rectangle is given by: Expanding this expression:
Step 3: Verifying the Options
The correct expression for the area as a function of is: Thus, the correct answer is:
Do you want more details or have any questions?
Here are 5 related questions you might explore:
- What are the dimensions of the dog run if the length is 10 feet?
- How would the area formula change if Elissa had more fencing?
- Can you find the value of that maximizes the area of the dog run?
- What is the maximum possible area of the dog run using the given fencing?
- 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.
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Math Problem Analysis
Mathematical Concepts
Geometry
Algebra
Quadratic Functions
Formulas
Perimeter of a rectangle = 2(length + width)
Area of a rectangle = length * width
Expanded form: A(x) = 14x - x^2
Theorems
Basic perimeter theorem for rectangles
Suitable Grade Level
Grades 8-10
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