A fence is to be built to enclose a rectangular area of 280 square feet. The fence along three sides is to be made of material that costs 6 dollars per foot, and the material for the fourth side costs 15 dollars per foot. Find the length L and width W (with W<= L) of the enclosure that is most economical to construct.
To find the most economical way to construct the enclosure, we need to minimize the total cost of the fence.
Let's assume the length of the enclosure is L and the width is W, where W <= L.
The area of the rectangle is given as 280 square feet, so we can write the equation: L * W = 280.
We need to find the dimensions that minimize the cost of the fence. The cost of the three sides (two lengths and one width) is 6 dollars per foot, and the cost of the fourth side (the remaining width) is 15 dollars per foot.
Now, let's express the total cost of the fence in terms of L and W:
Total cost = Cost of three sides + Cost of fourth side
Total cost = (2L + W) * 6 + W * 15
Total cost = 12L + 6W + 15W
Total cost = 12L + 21W
Now, we can express one variable in terms of the other using the area equation:
W = 280 / L
Substituting this into the total cost equation, we get:
Total cost = 12L + 21(280 / L)
To minimize the total cost, we need to differentiate this equation with respect to L and set it to zero:
d(Total cost) / dL = 12 - (5880 / L^2) = 0
Solving this equation, we get:
12L^2 = 5880
L^2 = 490
L = √490 ≈ 22.14
Now we can find the corresponding width using the area equation:
W = 280 / L ≈ 280 / 22.14 ≈ 12.64
Therefore, the length L is approximately 22.14 feet, and the width W is approximately 12.64 feet.