Question

A farmer has 900 feet of fence to build a pen that is up against an existing wall, so fencing is only needed on three sides. What will be the dimensions of the pen that will maximize the area?

Answer 1

The farmer has 900 feet of fencing and wants to construct a pen that is abutting an existing wall. The fence will be used on the other three sides, with two sides being equal in length (we'll call the length of these sides x) and the base of the rectangle representing the other side (we'll call this y).

The total length of fencing is represented as: 2x + y = 900

The area of the rectangle can be calculated as: A = xy.

We want to find the maximum area, so we can substitute y from our first equation (total fencing) into the area equation, giving us:

A = x(900 - 2x)

To find the maximum area, we calculate the derivative of A with respect to x, equate it to zero and solve for x. The derivative is given by:

A' = 900 - 4x

Setting this to zero, we get:

900 - 4x = 0

Solving for x, we find:

x = 900 / 4 = 225 feet

Now we substitute x back into our first equation (total fencing), to find the length of the base, y:

2*225 + y = 900

Solving for y, we find:

y = 900 - 450 = 450 feet

Therefore, to maximize the area, the pen should have its two equal sides each be 225 feet and the base be 450 feet. This configuration gives the maximum possible area for the pen.

Answer 2

Assuming a rectangular fenced area, let the area measure
`x` feet by
`y` feet. Then the length of fencing used is such that

`2x + y = 900 \implies y = 900 - 2x`

where
`y` is the length of the side parallel to the existing wall.

The area enclosed by the fence is

`A(x,y) = xy \implies A(x) = x(900-2x) = 900x - 2x^2`

Find the critical points of
`A(x)`.

`A'(x) = 900 - 4x = 0 \implies x = 225`

If two sides measure 225 feet, then the remaining side must measure

`y = 900 - 2\cdot225 = 450`

feet. So the pen must have dimensions 225-ft by 450-ft. We know this gives a maximum area, since by completing the square we find an upper bound for the enclosed area of

`900x - 2x^2 = -2 (x^2 - 450x) \\\\ ~~~~~~~~ = -2 (x^2 - 450x + 225^2 - 225^2) \\\\ ~~~~~~~~ = 2\cdot225^2 - 2 (x - 225)^2 \\\\ ~~~~~~~~ \le 2\cdot225^2 = 101,250`

square feet.