All 50 states have legislation regarding pool safely, the majority of which includes a requirement for safely fence around the perimeter of the top of an in-ground pool. If a rectangular in-ground pool has a length that is 2 feet less than twice its width, and the area of the top of the pool is 480 square feet, how many linear feet of fencing are required, assuming the fence is placed 1 foot from the edge of the water on all sides?

Options

A. `92`
B. `100`
C. `108`
D. `116`

Explanation

To solve the problem step by step, we will follow the instructions given in the question.

### Step 1: Define the Variables
Let the width of the pool be \( w \) feet. According to the problem, the length \( l \) of the pool is given as:
\[ l = 2w - 2 \]

### Step 2: Set Up the Area Equation
The area \( A \) of a rectangle is calculated as:
\[ A = \text{length} \times \text{width} \]
Given that the area of the pool is 480 square feet, we can set up the equation:
\[ w \times (2w - 2) = 480 \]

### Step 3: Expand and Rearrange the Equation
Expanding the left side, we get:
\[ 2w^2 - 2w = 480 \]
Now, rearranging this equation gives:
\[ 2w^2 - 2w - 480 = 0 \]
Dividing the entire equation by 2 simplifies it to:
\[ w^2 - w - 240 = 0 \]

### Step 4: Solve the Quadratic Equation
To solve the quadratic equation \( w^2 - w - 240 = 0 \), we can use the quadratic formula:
\[ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -1 \), and \( c = -240 \). Plugging in these values:
\[ w = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-240)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ w = \frac{1 \pm \sqrt{1 + 960}}{2} \]
\[ w = \frac{1 \pm \sqrt{961}}{2} \]
\[ w = \frac{1 \pm 31}{2} \]
This gives us two potential solutions:
\[ w = \frac{32}{2} = 16 \quad \text{and} \quad w = \frac{-30}{2} = -15 \]
Since width cannot be negative, we take \( w = 16 \) feet.

### Step 5: Calculate the Length
Now we can find the length \( l \):
\[ l = 2w - 2 = 2(16) - 2 = 32 - 2 = 30 \text{ feet} \]

### Step 6: Calculate the Perimeter for the Fencing
The perimeter \( P \) of the pool is given by:
\[ P = 2(l + w) = 2(30 + 16) = 2(46) = 92 \text{ feet} \]
However, since the fence is placed 1 foot from the edge of the water on all sides, we need to add 2 feet to both dimensions (1 foot on each side):
\[ \text{New width} = 16 + 2 = 18 \text{ feet} \]
\[ \text{New length} = 30 + 2 = 32 \text{ feet} \]
Now, calculate the new perimeter:
\[ P_{\text{fence}} = 2(18 + 32) = 2(50) = 100 \text{ feet} \]

### Final Answer
The total linear feet of fencing required is:
\[ \boxed{100} \]