Find the maximum area of the rectangular garden.

Question

A gardener is fencing a rectangular garden with 112 feet of fence. Find the maximum area of the rectangular garden.

Answer 1

112 feet of fence is the length of the fencing.  The area of a rectangle is length times width.  The perimeter of a rectangle is the sum of twice the length and twice the width.

 

Let length = x

Let width = y

 

Use these variables to create equations.

 

2(x + y) = 112              eq1

 

x + y = 56                    eq1

 

Area = xy                       eq2

 

 

Substitute eq1 into eq2 to get a function area in terms of only x and y.

 

Area = x(56 - x)

 

Area = -x² + 56x

 

 

The area represents a parabola that opens downward.  What you need to do is find the vertex of this parabola.

 

Find the x-coordinate of the vertex using this formula:

 

x = -b / (2a)

 

where:

a = -1

b = 56

 

Plug in these values into the formula.  Then evaluate the area at that x-coordinate.    That will be your maximum area. 

Answer 2

The maximum area occurs when the rectangular garden is a square. The area of a rectangular field is the length times the width, which we can call xy, where x is the width and y is the length. We have 112 ft of fence, so the perimeter P is given by:

 

P = 2x + 2y = 112 or x + y = 56 when we divide by 2.

 

Let's solve this for y.

 

y = 56 - x

 

Therefore, the max area is given when A = xy = x(56 - x) = 56x - x²