Final answer:
To minimize the amount of fencing needed to create an area that is 22,500 square feet and divide it in half, we can use the concept of optimization. The dimensions of the field that will minimize the perimeter can be found by solving for x and y in the equation xy = 22,500 and then calculating the actual perimeter using the obtained dimensions.
Step-by-step explanation:
To minimize the amount of fencing needed to create an area that is 22,500 square feet and divide it in half, we can use the concept of optimization. Let the length of the field be represented by x and the width be represented by y. We need to find the dimensions of the field that will minimize the perimeter. The perimeter is given by the formula P = 2x + 2y. Since we want to divide the area in half, we have the equation xy = 22,500. We can then solve for y in terms of x by rearranging the equation as y = 22,500/x. Substituting this into the perimeter formula, we get P = 2x + 2(22,500/x). To find the dimensions that minimize the perimeter, we can take the derivative of the perimeter equation with respect to x, set it equal to zero, and solve for x. The critical points will give us the possible values of x. Once we find the values of x, we can substitute them back into the equation to find the corresponding values of y. Finally, we can calculate the actual perimeter using the dimensions obtained.