A farmer has 90 meters of fencing to use to create a rectangular garden in the middle of an open field.

Question

A farmer has 90 meters of fencing to use to create a rectangular garden in the middle of an open field.

Write a formula that expresses  in terms of .

What is the maximum area of the garden?

What is the length and width of the garden that produces the maximum area?

1. 
   

If the farmer instead has 260 meters of fencing to enclose the garden, what fence-length and fence-width will produce the garden with the maximum area?

Answer 1

The two needed equations are for rectanglular perimeter and area. ( P for Perimeter; A for Area; W for Width and L for Length).

P = 2W + 2 L and A = W X L

Since P = 90 then 90 = 2 L + 2 W

Solve for W by subtracting 2 L from each side: 90-2L = 2W

Divide both sides for 2: 45 - L = W.

Now substitute this equation in for W in the area equation: (45-L) X L = A

Multiply through: 45L - L^2 = A rewrite into a standard parabola format: A = -L^2 + 45 L ( Part A)

This is the equation of a parabola and the maximum of the parabola with be maximum area of the garden.

The standard form of a parabola equation is y = ax^2 + bx + c. The vertex of a parabola is where the x term is - b/2a or for this example: -45/2(-1) or 45/2 = 22.5. Subituting 22.5 back into the equation :

area at maximum = -(22.5)^2 + 45 (22.5) = 22.5( -22.5+ 45) = 22.5^2 or 506.25 (Part B)

The length and width of the garden at maximum area if 22.5 feet (Part C).

As shown in the first example a square garden has the maximum area.( As the length increases the garden becomes more narrow and has less area.)

for 260 feet of fencing the sides of the square will be 260/4 or 65 feet each.