Question

A rectangular park has sidewalks around the perimeter. The park is 102 yards long and

94 yards wide. If you are at one corner and want to walk to get to the opposite corner,
you can either walk around on the sidewalk, or cut through the park diagonally as a
shortcut. How much longer is the path on the sidewalk than the shortcut?
57.3 yards
98.3 yards
156.4 yards
62.1 yards

Answer 1

The path on the sidewalk around the rectangular park is approximately 253.29 yards longer than the shortcut diagonal. None of the provided options exactly match this result, but 156.4 yards is the closest.

To determine the length of the path on the sidewalk, we need to find the perimeter of the rectangular park. The perimeter (\(P\)) of a rectangle is given by the formula:

`\[ P = 2 * (\text{length} + \text{width}) \]`

Given that the length of the park is 102 yards and the width is 94 yards:

`\[ P = 2 * (102 + 94) = 2 * 196 = 392 \, \text{yards} \]`

Now, let's find the length of the diagonal (shortcut) using the Pythagorean theorem, where \(d\) is the diagonal, \(l\) is the length, and \(w\) is the width:

`\[ d = √(l^2 + w^2) \]`

`\[ d = √(102^2 + 94^2) \]`

`\[ d = √(10404 + 8836) \]`

`\[ d = √(19240) \]`

`\[ d \approx 138.71 \, \text{yards} \]`

Now, we can find how much longer the path on the sidewalk is compared to the shortcut:

`\[ \text{Difference} = P - d \]`

`\[ \text{Difference} = 392 - 138.71 \]`

`\[ \text{Difference} \approx 253.29 \, \text{yards} \]`

Therefore, the path on the sidewalk is approximately 253.29 yards longer than the shortcut. None of the provided options exactly match this result, but the closest option is 156.4 yards. However, none of the provided options seem to be accurate, so there might be an error in the options.