A fence is to be built to enclose a rectangular area of 220 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 13 dollars per foot. Find the dimensions of the enclosure that is most economical to construct.

To find the dimensions of the enclosure that is most economical to construct, we need to minimize the cost of the fence.

Let's assume the length of the rectangular area is L and the width is W.

1. The fence along three sides: The length of two sides and one width.
The cost of the fence along these three sides = 2L + W

2. The fence along the fourth side: The width of the rectangle.
The cost of the fence along the fourth side = W

3. The total cost of the fence = Cost of three sides + Cost of the fourth side
Total Cost = 2L + W + W * 13

4. The area of the rectangle = Length * Width
220 = L * W

Now, we can express the cost (Total Cost) in terms of a single variable, for example, L:

Total Cost = 2L + W + W * 13
Total Cost = 2L + (220 / L) + (220 / L) * 13
Total Cost = 2L + 220/L + 2860/L

To minimize the cost, we need to find the value of L that gives the minimum value of Total Cost. To do this, we can take the derivative of Total Cost with respect to L and set it equal to zero:

d(Total Cost) / dL = 2 - 220/L^2 - 2860/L^2 = 0

Simplifying the equation:
2L^2 - 220 - 2860 = 0
2L^2 - 3080 = 0

Now, we can solve for L by using the quadratic formula:
L = (-b ± sqrt(b^2 - 4ac)) / (2a)

where a = 2, b = 0, and c = -3080.

L = (-0 ± sqrt(0^2 - 4 * 2 * -3080)) / (2 * 2)
L = (± sqrt(0 + 24640)) / 4
L = (± sqrt(24640)) / 4
L ≈ ± 78.78 / 4
L ≈ ± 19.7

Since the length of a fence cannot be negative, we ignore the negative solution, and we have L ≈ 19.7.

Now, we can find the width (W) using the equation for the area of the rectangle:
220 = L * W
220 = 19.7 * W
W ≈ 220 / 19.7
W ≈ 11.17

Therefore, the dimensions of the enclosure that is most economical to construct are approximately L ≈ 19.7 feet and W ≈ 11.17 feet.

To find the dimensions of the enclosure that is most economical to construct, we need to consider the cost of the fence and find the dimensions that minimize the cost.

Let's assume the length of the rectangular area is L and the width is W.
The area of the rectangular area is given as 220 square feet, so we have the equation:
L * W = 220

The cost of the fence along three sides is $6 per foot, which means the total cost for these three sides is:
Cost_three_sides = 2 * (L + W) * 6

The cost of the fence along the fourth side is $13 per foot, which means the cost for this side is:
Cost_fourth_side = L * 13

The total cost of the fence can be expressed as the sum of the costs of the three sides and the fourth side:
Total_cost = Cost_three_sides + Cost_fourth_side

Substituting the expressions for the costs, we get:
Total_cost = 2 * (L + W) * 6 + L * 13

To find the dimensions that minimize the cost, we can differentiate the total cost with respect to L and W, set the derivatives equal to zero, and solve for L and W.

d(Total_cost) / dL = 0
d(Total_cost) / dW = 0

Differentiating the total cost equation with respect to L and W, we get:
12 + 13 = 0 (1)
12 + 13 = 0 (2)

Now, let's solve equations (1) and (2) simultaneously to find L and W.

From equation (1), we have:
12 + 13 = 0
12 = -13
This equation doesn't hold true, so we can disregard it.

From equation (2), we have:
12 + 13 = 0
25 = 0
This equation also doesn't hold true, so we can disregard it.

Therefore, there seems to be an error or inconsistency in the problem statement or calculations. Please double-check the information provided.

a and b are lengths

A = 220 = a * b so a = 220/b

cost = 2(6a) + 1 (6b) + 1(13b)
cost = c = 12 a + 19 b
c = 12(220/b)+19 b
dc/db = 0 at max or min = 12(-220/b^2) + 19

220(12) = 19 b^2
solve for b and a