What is the answer to this?? I don't get it

Question

The question was A farmer has 1756 feet of fencing available to enclose a rectangular area bordering a river. If no fencing is required along the river, find the dimensions of the fenced area that will maximize the area. What is the maximum area?

Answer 1

From the above answer (from Raymond B.) you have the quadratic function for the Area:

A(w) = 1756w - 2w²

His solution uses derivatives. If you're not familiar with the derivatives, here is another solution.

After slight rearrangement we have

A(w) = -2w² + 1756w

This is a parabola, opening down since the coefficient to w² is negative (-2). It means that the vertex of this parabola is the maximum of the function, and this is exactly what we need.

The x-coordinate of the vertex (in our case it's w-variable) is:

w = -1756 / (2 x (-2)) = 1756 / 4 = 439 <--- this is the Width

And the Length = 1756 - 2w = 878

We can find the area in two ways:

1. Width by Length = 439 x 878 = 385442 or
2. A(439) = -2w² + 1756w² = -2 x (439)² + 1756 x 439 = 385442

Hope this helps

Answer 2

1756/2 = 878 feet = Length

878/2 = 439 feet = Width

2 sides are Widths

1 side is the Length

the Length is twice the size of the Width

max Area = LW = 878(439) =385,442 square feet

or use differential calculus

Perimeter = 1756 = L+2W

L=1756 -2W

A=LW = (1756-2W)W= 1756W -2W^2

take the derivative and set equal to zero

A' = -4W + 1756 = 0

W =1756/4 = 439 feet Wide

L= 1756-2(439) = 878 feet Long