A farmer decides to enclose a rectangular​ garden, using the side of a barn as one side of the rectangle. what is the maximum area that the farmer can enclose with 100 ft of​ fence? what should the dimensions of the garden be to give this​ area?

The perimeter will be:
 P = 2x + y
 100 = 2x + y
 The area is:
 A = x * y
 We write the area as a function of x:
 A (x) = x * (100-2x)
 Rewriting:
 A (x) = 100x - 2x ^ 2
 We derive:
 A '(x) = 100 - 4x
 We equal zero and clear x:
 0 = 100 - 4x
 4x = 100
 x = 25 feet
 The other dimension is:
 y = 100-2x
 y = 100-2 (25)
 y = 100-50
 y = 50 feet
 The area will be:
 A = (25) * (50)
 A = 1250 feet ^ 2
 Answer:
 the maximum area that the farmer can enclose with 100 ft of fence is:
 A = 1250 feet ^ 2
 The dimensions of the garden to give this area should be:
 x = 25 feet
 y = 50 feet