You have 7L feet of fence to make a rectangular vegetable garden alongside the wall of your house, where L is a positive constant. The wall of the house bounds one side of the vegetable garden. What is the largest possible area for the vegetable garden?

Answer 1

Let the side parallel to the house-wall by y
let the other two sides of the rectangle be x
so y + 2x = 2L
y = 2L - 2x

area = xy = x(2L - 2x) = -2x^2 + 2Lx

the maximum area occurs at the vertex of this downward parabola
the x of the vertex is -b/(2a) = -2L/-4 = L/2
so that max is -2(L/2)^2 + 2L(L/2) = .....

Answer 2

i