Final answer:
For the most economical fence enclosing an area of 1800 square feet with different costs per side, we want the cheaper side to be longer. The dimensions that satisfy this, while fitting the area requirement, are a short side of 40 ft and a long side of 45 ft (Option B).
Step-by-step explanation:
The objective is to find the dimensions of the rectangle that result in the most economical fence. We know the area of the rectangle is 1800 square feet, and we can represent the length and the width of the rectangle as L and W respectively. The area is given by the equation L x W = 1800. We also know that three sides of the fence cost $5 per foot and one costs $15 per foot. Therefore, the total cost of the fence would be Cost = 5(2W + L) + 15L.
Using the area equation L x W = 1800, we can express one variable in terms of the other, for instance, L = 1800/W. Substituting this into the cost equation gives Cost = 5(2W + 1800/W) + 15(1800/W). To minimize the cost, we can take the derivative with respect to W and find the critical points. The dimensions that result in the least cost will be the valid critical points within the context of the problem.
Choosing between the given options, we want the side with the cheaper material to be longer to minimize the cost. Option B, with dimensions Short side: 40 ft, Long side: 45 ft, meets the requirements of the area (40 ft x 45 ft = 1800 sq ft) and represents the most economical configuration in comparison to the other options.