To find the value of \( x \) in the rectangle's side lengths, we first need to use the formula for the perimeter of a rectangle. The perimeter \( P \) of a rectangle is given by:
\[ P = 2 \times (\text{length} + \text{width}) \]
Given the side lengths:
- Length \( = 5x + 20 \) ft
- Width \( = 2x + 23 \) ft
We can plug these into the perimeter formula:
\[ 208.5 = 2 \times ((5x + 20) + (2x + 23)) \]
Simplifying inside the parentheses:
\[ 208.5 = 2 \times (5x + 20 + 2x + 23) \] \[ 208.5 = 2 \times (7x + 43) \]
Next, we'll distribute the 2:
\[ 208.5 = 14x + 86 \]
Now, we solve for \( x \) by isolating \( x \). First, subtract 86 from both sides:
\[ 208.5 - 86 = 14x \] \[ 122.5 = 14x \]
Next, divide by 14:
\[ x = \frac{122.5}{14} \] \[ x = 8.75 \]
Thus, the value of \( x \) is \( \boxed{8.75} \).