A rectangular playground is to be fenced off and divided into two parts by a fence parallel to one side of the playground. 840 feet of fencing is used. Find the dimensions of the playground that will enclose the greatest total area.

The dimensions of the playground that will enclose the greatest total area are 140 ft x 210 ft.

Step-by-step explanation:

We have a rectangular with sides "a" and "b", so that the area is:

The perimiter for this rectangle is

The fence is for the perimeter plus the division, which has a length of "a".

So the total fencing is:

We can express one side in function of the other, in order to optimize the area.

Then, we can write the area as:

To maximize the area we will derive and equal to zero

Then, the value for the other side of the rectangle is:

a reflection across the x-axis , followed by the by a reflection across the y-axis , and then a scale dilation of 0.5 .