A fence is to be built to enclose a rectangular area of 210 square feet. The fence along three sides is to be made of material that costs 4 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 solve this problem, we need to minimize the cost of the fence. The cost of the fence is determined by the perimeter of the rectangle, which is given by the formula P = 2l + 2w, where l is the length and w is the width of the rectangle. However, since one side of the fence costs more, we will denote the length of the rectangle as the side that costs $13 per foot, and the two widths and one length as the sides that cost$4 per foot.

The area of the rectangle is given by the formula A = lw, and we know that A = 210 square feet. We can solve this equation for l to get l = 210/w.

The cost of the fence is given by the formula C = 13l + 4(2w + l). Substituting the expression for l from above, we get C = 13(210/w) + 4(2w + 210/w).

To find the minimum cost, we need to find the minimum of this function. We can do this by taking the derivative of the cost function with respect to w, setting it equal to zero, and solving for w.

The derivative of the cost function is C' = -2730/w^2 + 8. Setting this equal to zero gives -2730/w^2 + 8 = 0. Solving for w gives w = sqrt(2730/8) = 18.41 feet.

Substituting w = 18.41 feet back into the equation for l gives l = 210/18.41 = 11.4 feet.

Therefore, the dimensions that will minimize the cost of the fence are a length of 11.4 feet and a width of 18.41 feet.