To find the total distance walked by the person along the sidewalks outside the park, we can use trigonometric functions involving the right triangle formed by the diagonal path, the long side of the park, and the sidewalk.
Given:
- The length of the diagonal (hypotenuse) = 100 yards
- Angle adjacent to the hypotenuse = 40 degrees
In this right triangle, we can use the following trigonometric relationships:
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The adjacent side \( X \) (the length of the long side of the park) can be determined using the cosine function: \[ X = 100 \cdot \cos(40^\circ) \]
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The opposite side \( Y \) (the length of the short side of the park) can be calculated using the sine function: \[ Y = 100 \cdot \sin(40^\circ) \]
However, we need to find the distance walked along the sidewalks, which will consist of both sides \( X \) and \( Y \):
The total distance \( D \) walked along the sidewalks outside the park is: \[ D = X + Y \]
Substituting the expressions for \( X \) and \( Y \): \[ D = 100 \cdot \cos(40^\circ) + 100 \cdot \sin(40^\circ) \]
This can be factored out: \[ D = 100 \cdot (\cos(40^\circ) + \sin(40^\circ)) \]
Thus, the expression showing the distance that he walked is: \[ \boxed{100 \cdot (\cos(40^\circ) + \sin(40^\circ))} \]