A man has 400 meters of fencing and wants to fence in a rectangular field next to a straight river. Assuming that no fencing is needed along the river, what are the dimensions of the field with the largest possible area?

The largest area would be a square, or as close to a square as you can get, 400/3 by 400/3.

To find the dimensions of the field with the largest possible area, we can use the technique of optimization. The objective is to maximize the area of the field while using a fixed amount of fencing (400 meters in this case).

Let's assume the length of the field is 'L' and the width is 'W'. Since no fencing is needed along the river, the length 'L' of the field will be the side perpendicular to the river, and the width 'W' will be the side parallel to the river.

Given that the total fencing available is 400 meters, we can express the perimeter of the field as:

Perimeter = 2L + W = 400

Now, we can isolate 'W' in terms of 'L' by rearranging the equation:

W = 400 - 2L

The area of a rectangle is given by:

Area = Length * Width = L * W

Substituting the value of 'W' from the perimeter equation, we get:

Area = L * (400 - 2L)

To find the dimensions that maximize the area, we need to find the value of 'L' that maximizes this formula. We can do this by taking the derivative of the area formula with respect to 'L', setting it equal to zero, and solving for 'L'.

Differentiating the area formula, we get:

d(Area)/dL = 400 - 4L

Setting this derivative equal to zero, we have:

400 - 4L = 0

Solving this equation for 'L', we find:

L = 100

Now that we have the value of 'L', we can substitute it back into the perimeter equation to find the corresponding value of 'W':

W = 400 - 2(L) = 400 - 2(100) = 200

Therefore, the dimensions of the field with the largest possible area are L = 100 meters and W = 200 meters.