To find the length of the diagonal path in a rectangular park, we can use the Pythagorean theorem. The theorem states that for a right triangle, the square of the length of the hypotenuse (in this case, the diagonal path) is equal to the sum of the squares of the lengths of the other two sides (the length and width of the rectangle).
Let \( a = 45 \) yards and \( b = 31 \) yards.
The formula for the diagonal \( d \) is given by:
\[ d = \sqrt{a^2 + b^2} \]
Substituting the values of \( a \) and \( b \):
\[ d = \sqrt{45^2 + 31^2} \]
Calculating \( 45^2 \) and \( 31^2 \):
\[ 45^2 = 2025 \] \[ 31^2 = 961 \]
Now, add these two results:
\[ 2025 + 961 = 2986 \]
Next, take the square root:
\[ d = \sqrt{2986} \]
Calculating the square root of \( 2986 \):
\[ d \approx 54.7 \]
Rounding to the nearest tenth, the length of the diagonal path is approximately \( \boxed{54.7} \) yards.
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