Final answer:
To find the greatest area someone can enclose with a given budget, we need to optimize the dimensions of the rectangular area. By using the perimeter equation and maximizing the product of the length and width, we find that the maximum area is approximately 3164.06 m².
Step-by-step explanation:
To find the greatest area someone can enclose with a given budget, we need to optimize the dimensions of the rectangular area. Let's assume the length of the rectangular area is L and the width is W. The cost of the border edge is $20 per meter, so the perimeter of the rectangular area would be P = 2(L + W). We know that the total cost is $4500, so we can write the equation 20P = 4500.
Substituting P with the expression 2(L + W), we have 20(2L + 2W) = 4500. Simplifying the equation, we get 4L + 4W = 225. Rearranging the equation, we get L = (225 - 4W)/4.
To maximize the area, we need to maximize the product L * W. By substituting the expression for L from the previous equation, we have (225 - 4W)(W) = 225W - 4W^2. To find the maximum value of this quadratic equation, we can take the derivative with respect to W, set it equal to 0, and solve for W. The resulting value of W can then be substituted back into the equation for L to find the corresponding value of L. The maximum area would be given by L * W.
After solving the equations, we find that the maximum area is approximately 3164.06 m².