Questions: A protected wilderness area in the shape of a rectangle is 3 kilometers long and 3.3 kilometers wide. The forest is to be surrounded by a hiking trail that will cost 11,000 per mile to construct. What will it cost to install the trail? Note that 1 mile (mi) ≈ 1.6 kilometers (km).

Solution Steps

To solve this problem, we need to calculate the perimeter of the rectangular wilderness area, convert the perimeter from kilometers to miles, and then determine the cost of constructing the hiking trail based on the given cost per mile.

1. Calculate the perimeter of the rectangle.
2. Convert the perimeter from kilometers to miles.
3. Multiply the perimeter in miles by the cost per mile to get the total cost.

The perimeter \( P \) of a rectangle is given by the formula: \[ P = 2 \times (L + W) \] where \( L = 3 \, \text{km} \) and \( W = 3.3 \, \text{km} \). Thus, \[ P = 2 \times (3 + 3.3) = 2 \times 6.3 = 12.6 \, \text{km} \]

To convert kilometers to miles, we use the conversion factor \( 1 \, \text{km} \approx 0.625 \, \text{mi} \). Therefore, the perimeter in miles is: \[ P_{\text{miles}} = 12.6 \, \text{km} \times 0.625 \, \text{mi/km} = 7.875 \, \text{mi} \]

The total cost \( C \) to construct the trail is calculated by multiplying the perimeter in miles by the cost per mile: \[ C = P_{\text{miles}} \times \text{cost per mile} = 7.875 \, \text{mi} \times 11000 \, \text{\$} = 86625.0 \, \text{\$} \]

Final Answer