To find the maximum area that the farmer can enclose with 100 ft of fence, we need to consider the dimensions of the rectangle.
Let's assume the length of the rectangle is "L" and the width is "W". Since the farmer is using the side of a barn as one side of the rectangle, we only need to fence the remaining three sides.
Therefore, the perimeter of the rectangle can be written as:
Perimeter = L + 2W
According to the question, the perimeter is given as 100 ft:
100 = L + 2W
We can solve this equation to find an expression for L in terms of W:
L = 100 - 2W
Now, let's calculate the area of the rectangle:
Area = L * W
Substituting the expression for L:
Area = (100 - 2W) * W
To find the maximum area, we can take the derivative of the area function with respect to W and set it equal to zero to find the critical point.
Let's differentiate the area function:
d(Area)/dW = 100 - 4W
Setting the derivative equal to zero:
100 - 4W = 0
4W = 100
W = 25 ft
Now, we can substitute the value of W back into the expression for L to find its value:
L = 100 - 2W = 100 - 2(25) = 50 ft
Therefore, the maximum area that the farmer can enclose with 100 ft of fence is:
Area = L * W = 50 ft * 25 ft = 1250 sq ft
The dimensions of the garden to give this area are:
50 ft by 25 ft.