Final answer:
The dimensions of the rectangular field of maximum area that can be enclosed with 240 m of fencing, with no fencing needed on one side, are 60 meters by 120 meters.
Step-by-step explanation:
To find the dimensions of a rectangular field of maximum area, we can use the formula for the area of a rectangle, which is length multiplied by width. Let's assume that the length of the field is x meters. Since there is no fencing needed on one side, the width of the field will be 240 - 2x meters. Therefore, the area of the field can be expressed as A = x(240 - 2x).
To find the maximum area, we can take the derivative of the area function, set it equal to zero, and solve for x. The critical point we find will give us the length of the field that maximizes the area. Taking the derivative of the area function, we get dA/dx = 240 - 4x.
Setting dA/dx equal to zero and solving for x, we get 240 - 4x = 0, which gives us x = 60. Therefore, the length of the field that maximizes the area is 60 meters, and the width is 240 - 2(60) = 120 meters. So, the dimensions of the rectangular field of maximum area are 60 meters by 120 meters.