a farmer wants to enclose a rectangular area along a highway. the fencing along the highway will be made of metal costing $20/ft the rest will be made of wood costing $5/ft what are the dimensions that will minimize the cost to fence a field with an area of 1600 ft

To minimize the cost of fencing a rectangular field with an area of 1600 square feet, you can use calculus. Let's denote the dimensions of the field as follows: Length of the field: L (in feet) Width of the field: W (in feet) The area of the field is given as 1600 square feet, so we have the equation: L * W = 1600 The cost of the metal fencing along the highway (two sides) is $20 per foot, and the cost of the wood fencing (two sides) is $5 per foot. Therefore, the total cost C can be expressed as: C = 20(L) + 5(L) + 5(W) + 5(W) = 25(L + W) Now, we want to minimize C subject to the constraint L * W = 1600. We can use the constraint to express one variable in terms of the other. For example, we can express L as: L = 1600 / W Now, substitute this into the cost equation: C = 25(1600 / W + W) Next, take the derivative of C with respect to W, set it equal to zero to find critical points, and then determine which critical point minimizes C. dC/dW = 25(-1600/W^2 + 1) = 0 Solve for W: -1600/W^2 + 1 = 0 -1600/W^2 = -1 W^2 = 1600 W = ±40 Since width cannot be negative in this context, we take W = 40 feet. Now, use this value to find L: L = 1600 / 40 = 40 feet So, the dimensions that will minimize the cost to fence the field with an area of 1600 square feet are: Length (L) = 40 feet Width (W) = 40 feet This will minimize the cost of fencing while meeting the area requirement.