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 would be:
We need to solve the
perimeter formula for either
l or
w. Let’s solve for
w:
Now substituting
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:
The
graph of
will be a
parabola and, since
, 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 for
area yields:
Therefore the largest
area that the farmer could enclose would be a
square where each
side has a
length 250 yards.