Let the length of the patio be \( L \) feet. According to the problem, the width \( W \) of the patio is given by:
\[ W = 2L - 40.75 \]
The perimeter \( P \) of the rectangle is given by the formula:
\[ P = 2L + 2W \]
We know that the perimeter is \( 86.5 \) feet, so we can set up the equation:
\[ 2L + 2W = 86.5 \]
Substituting the expression for \( W \):
\[ 2L + 2(2L - 40.75) = 86.5 \]
Now simplify the equation:
\[ 2L + 4L - 81.5 = 86.5 \]
Combine like terms:
\[ 6L - 81.5 = 86.5 \]
Now isolate \( 6L \):
\[ 6L = 86.5 + 81.5 \]
\[ 6L = 168 \]
Now divide both sides by 6 to find \( L \):
\[ L = \frac{168}{6} = 28 \]
So, the length of the patio is \( \boxed{28} \) feet.
To confirm, let's calculate the width using the expression for \( W \):
\[ W = 2(28) - 40.75 = 56 - 40.75 = 15.25 \]
Now, check the perimeter:
\[ P = 2L + 2W = 2(28) + 2(15.25) = 56 + 30.5 = 86.5 \]
This confirms that our calculations are correct. The length of the patio is indeed \( \boxed{28} \) feet.
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