Question

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?

Answer 1

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