You have 25-1.5foot fence pieces and 1--1.5foot gate. If you place the fenced zone against a barn of infinite size, what is the maximum possible (rectangular) area in the fenced zone, and how long are the sides?

To find the maximum possible area in the fenced zone, we need to determine the dimensions that maximize the area of a rectangle using the given fence pieces.

Let's start by calculating the length of the sides of the rectangle. Each fence piece has a length of 1.5 feet, and we have a total of 25 fence pieces. So, the combined length of all the fence pieces is 1.5 * 25 = 37.5 feet.

Since we also have a gate which takes up a 1.5-foot section, we need to subtract this from the combined length of the fence pieces. Therefore, the usable length for the sides of the rectangle is 37.5 - 1.5 = 36 feet.

Now, let's find the dimensions that maximize the area. Since the area of a rectangle is maximized when it has equal side lengths (a square), we need to find two side lengths that add up to 36 feet (the usable length).

We can try different combinations of side lengths that add up to 36 to find the dimensions that maximize the area.

Let's start with a square. If both sides have a length of 18 feet, then the area would be 18 * 18 = 324 square feet.

Now, let's try a rectangle. We can choose a different length for one side while keeping the sum of the sides equal to 36.

Let's say one side has a length of 20 feet; then the other side would need to be 16 feet to maintain a sum of 36.

The area of this rectangle would be 20 * 16 = 320 square feet.

We can continue trying different dimensions, but we can see that the maximum area is obtained when the rectangle is actually a square, with side lengths of 18 feet each.

Therefore, the maximum possible rectangular area in the fenced zone is 324 square feet, and the sides are 18 feet long.