To solve the problem step by step, we will follow the given information and derive the necessary values to find the perimeter of Jonathon's fence.
### Step 1: Define the Variables
Let:
- \( L \) = Length of the fence (in inches)
- \( W \) = Width of the fence (in inches)
### Step 2: Set Up the Relationship Between Width and Length
According to the problem, the width is given as:
\[ W = 5L + 350 \]
### Step 3: Use the Area Formula
The area of a rectangle is given by the formula:
\[ \text{Area} = L \times W \]
We know the area is 64,680 square inches, so we can write:
\[ L \times W = 64,680 \]
### Step 4: Substitute the Width in the Area Equation
Substituting \( W \) from Step 2 into the area equation:
\[ L \times (5L + 350) = 64,680 \]
### Step 5: Expand the Equation
Expanding the left side:
\[ 5L^2 + 350L = 64,680 \]
### Step 6: Rearrange the Equation
Rearranging gives us:
\[ 5L^2 + 350L - 64,680 = 0 \]
### Step 7: Simplify the Equation
To simplify, we can divide the entire equation by 5:
\[ L^2 + 70L - 12,936 = 0 \]
### Step 8: Apply the Quadratic Formula
The quadratic formula is:
\[ L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( a = 1, b = 70, c = -12,936 \):
- Calculate \( b^2 - 4ac \):
\[ 70^2 - 4 \times 1 \times (-12,936) = 4900 + 51,744 = 56,644 \]
Now apply the quadratic formula:
\[ L = \frac{-70 \pm \sqrt{56,644}}{2 \times 1} \]
### Step 9: Calculate the Values
Calculating \( \sqrt{56,644} \) gives approximately \( 238.1 \):
\[ L = \frac{-70 \pm 238.1}{2} \]
### Step 10: Determine the Positive Length
Calculating the two possible values for \( L \):
1. \( L = \frac{-70 + 238.1}{2} \approx \frac{168.1}{2} \approx 84.05 \)
2. \( L = \frac{-70 - 238.1}{2} \) will give a negative value, which we discard.
Thus, we take:
\[ L \approx 84 \text{ inches} \]
### Step 11: Calculate the Width
Now substitute \( L \) back into the equation for \( W \):
\[ W = 5(84) + 350 = 420 + 350 = 770 \text{ inches} \]
### Step 12: Calculate the Perimeter
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(L + W) \]
Substituting the values of \( L \) and \( W \):
\[ P = 2(84 + 770) = 2(854) = 1708 \text{ inches} \]
### Final Answer
The perimeter of Jonathon's fence is **1708 inches**.
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