Steve plans to use 28 feet of fencing to enclose region of his yard for a pen for his pet rabbit. What is the area, in square feet, of the largest rectangular region Steve can enclose?

Options

A. 40
B. 45
C. 48
D. 49

Answer

To find the area of the largest rectangular region Steve can enclose with 28 feet of fencing, we can follow these steps:

### Step 1: Understand the Perimeter Formula
The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2L + 2W \]
where \( L \) is the length and \( W \) is the width of the rectangle.

### Step 2: Set Up the Equation
Since Steve has 28 feet of fencing, we can set up the equation:
\[ 2L + 2W = 28 \]
Dividing the entire equation by 2 gives:
\[ L + W = 14 \]

### Step 3: Express Width in Terms of Length
From the equation \( L + W = 14 \), we can express \( W \) in terms of \( L \):
\[ W = 14 - L \]

### Step 4: Write the Area Formula
The area \( A \) of the rectangle is given by:
\[ A = L \times W \]
Substituting \( W \) from the previous step, we get:
\[ A = L \times (14 - L) = 14L - L^2 \]

### Step 5: Identify the Quadratic Function
The area function \( A = 14L - L^2 \) is a quadratic equation in the standard form \( A = -L^2 + 14L \). This is a downward-opening parabola.

### Step 6: Find the Vertex
The maximum area occurs at the vertex of the parabola. The \( L \)-coordinate of the vertex can be found using the formula:
\[ L = -\frac{b}{2a} \]
where \( a = -1 \) and \( b = 14 \):
\[ L = -\frac{14}{2 \times -1} = \frac{14}{2} = 7 \]

### Step 7: Calculate the Width
Now that we have \( L = 7 \), we can find \( W \):
\[ W = 14 - L = 14 - 7 = 7 \]

### Step 8: Calculate the Maximum Area
Now we can calculate the maximum area:
\[ A = L \times W = 7 \times 7 = 49 \text{ square feet} \]

### Final Answer
The area of the largest rectangular region Steve can enclose is:
\[ \boxed{49} \text{ square feet} \]

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