rectangular park had a dirt path across its diagonal that was 100

Question

yards long. The diagonal and the long side of the park formed an angle that measured 40
.

A person walked along the sidewalks outside the park, from the start to the end of the path, as shown by the arrows.

The diagonal dirt path that cuts across the rectangular park and the sidewalks outside the park form a right triangle. The dirt path is the hypotenuse and it is labeled 100 yards. The start of the sidewalk is adjacent to the 40-degree angle and it is labeled X. The end of the path is opposite the 40-degree angle and it is labeled Y.

Which expression shows the distance that he walked?

Answer 1

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:

• The adjacent side \( X \) (the length of the long side of the park) can be determined using the cosine function: \[ X = 100 \cdot \cos(40^\circ) \]

• 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: \[ \boxed{100 \cdot (\cos(40^\circ) + \sin(40^\circ))} \]