Final answer:
To minimize the cost of the fence, the dimensions of the rectangular field should be x = 2(√12,000,000)m and y = √12,000,000m.
Step-by-step explanation:
To minimize the cost of the fence, we need to find the dimensions of the rectangular field that minimize the perimeter. Let's assume the length of the field is x meters and the width is y meters.
The area of the field is given as 24.0 million square meters, so we have the equation xy = 24,000,000. We are also told to divide the field in half with a fence parallel to one of the sides, which means the length of the field will be divided in half. So the new length becomes (x/2) meters.
The cost of the fence is determined by its perimeter. The perimeter of the field is given by the equation 2x + 2y. Since we want to minimize the cost, we want to minimize the perimeter. Substituting x/2 for x in the equation gives us the perimeter as (2(x/2) + 2y) = (x + 2y).
We can solve for x or y in terms of the other variable from the equations xy = 24,000,000 and x = 2y. By substituting x = 2y in the equation xy = 24,000,000, we get 2y^2 = 24,000,000. Solving for y, we find y^2 = 12,000,000 and y = √12,000,000.
Substituting y = √12,000,000 in the equation x = 2y, we get x = 2(√12,000,000). Thus, the lengths of the sides of the rectangular field that minimize the cost of the fence are x = 2(√12,000,000) meters and y = √12,000,000 meters.