Question

a rectangular park had a dirt path across its diagonal that was 100 yards long. the diagonal and the long side of the park formed an angle that measured 30°. a person walked along the sidewalks outside the park, from the start to the end of the path, as shown by the arrows. which expression shows the distance that he walked? 200tan 30° ≈ 115 yards 100tan 30° + 100tan 60° ≈ 231 yards 100cos 30° + 100sin 30° ≈ 137 yards 100cot 30° ≈ 173 yards

Explanation:

Step1: Find the short - side length ($x$)

In a right - triangle (half of the rectangle), using the cosine function. $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\theta = 30^{\circ}$ and the hypotenuse is 100 yards. So, $x = 100\cos30^{\circ}$.

Step2: Find the long - side length ($y$)

Using the sine function. $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 30^{\circ}$ and the hypotenuse is 100 yards. So, $y = 100\sin30^{\circ}$.

Step3: Calculate the walking distance

The person walked along two sides of the rectangle. The distance $d=x + y=100\cos30^{\circ}+100\sin30^{\circ}$.

Answer:

$100\cos30^{\circ}+100\sin30^{\circ}\approx137$ yards (the third option)