Problema Solution
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 provided by our tutors
let 'x' and 'y' represent the sides of the rectangular garden
y = one of the sides where the barn is
the farmer uses 100 ft to enclose the garden:
2x + y =100
y = 100 - 2x
the area of the garden is given by:
A = x*y
A = x(100 - 2x)
A = -2x^2 + 100x
we need to find the maximum of the parabolic function A = -2x^2 + 100x
since the quotient in front of x^2 is -1 < 0 the function has maximum in the vertex
A max = c - b^2/4a, where a = -2, b = 100, c = 0
A max = 0 - 100^2/(4*(-2))
A max = 1,250 ft^2 => x = -b/2a that is x = -100/(2*(-2)) = 25 ft
y = 100 - 2*25
y = 50 ft
the dimensions of the garden should be: 25 ft and 50 ft.
the maximum area that the farmer can enclose is: 1,250 ft^2.