Quadratic Equations are used to find maximums and minimums for rectangular regions. An example of this type of problem would occur when a person, with a specific amount of fencing, wants to find the largest rectangular area
that can be fenced off.
In order to solve quadratic equations involving maximums and minimums for rectangular regions, it is necessary to
Suppose a farmer has 1000 yards of fencing to enclose a rectangular field. What is the largest area
that the farmer can enclose?
The figure shown below illustrates the rectangular fence that is to be built. The fence will surround the rectangular area, and therefore, will create the perimeter
of the region.
The region inside the fence is described by area. An expression
for this figure's area
We need to solve the perimeter
formula for either l
. Let’s solve for w
into the area
formula we have:
Since A represents a quadratic equation
) in terms of l
, we will re-write A in function
form with the exponents in descending order:
will be a parabola
, the parabola
will have a maximum point
as its vertex. The y-coordinate of the vertex
will represent our greatest area. To proceed, we need to find the value of the x-coordinate of the vertex
(that is, the value of l
in our equation).
Substituting this value for l
into our equation
Therefore the largest area
that the farmer could enclose would be a square
where each side
has a length