a farmer is building a fence around his yard. one side of the fence will be the side of his barn , the other three sides will be made of wood. He has enough materials for 120m of fence, what is the maximum area he can enclose ? what are the dimensions which give this area

A square will give maximum area, 40m by 40m.

To find the maximum area the farmer can enclose with a given amount of fence, we need to consider the possible dimensions of the yard. Let's define the side of the barn as 'x' and the other two sides as 'y' and 'z'.

Since one side of the fence is the barn, it means that the remaining three sides, y, z, and x (barn side), must add up to a total of 120 meters.

The perimeter of a rectangle is given by the formula P = 2(length + width). In this case, the perimeter is 120 meters, so we can write the equation as:

2(x + y + z) = 120

Now, we need to express 'y' or 'z' in terms of 'x' so that we can write the area in terms of a single variable. Let's solve for 'y':

2(x + y + z) = 120
x + y + z = 60
y = 60 - x

The area of a rectangle is given by the formula A = length * width. In this case, the length is the side of the barn 'x', and the width is 'y'.

A = x * y
A = x * (60 - x)

To find the maximum area, we can use calculus. We need to find the critical points by taking the derivative of the area equation with respect to 'x' and setting it to zero:

dA/dx = 60 - 2x

Setting the derivative equal to zero:

60 - 2x = 0
2x = 60
x = 30

Now that we have the critical point, we can determine whether it is a maximum or minimum by taking the second derivative of the area equation:

d^2A/dx^2 = -2

Since the second derivative is negative, it indicates that the critical point is a maximum.

Therefore, the maximum area the farmer can enclose is when the side of the barn, 'x', is equal to 30 meters. To find the dimensions that give this maximum area, we substitute this value of 'x' back into the equation for 'y':

y = 60 - x
y = 60 - 30
y = 30

Therefore, the dimensions that give the maximum area are:
Length (side of the barn): 30 meters
Width (other two sides): 30 meters

This means the maximum area the farmer can enclose is 30 meters * 30 meters = 900 square meters.