Word Problems: Quadratic Max/Min Application  Rectangular Areas 
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 rewrite 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 ycoordinate of the vertex will represent our greatest area. To proceed, we need to find the value of the xcoordinate 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.

Let's Practice








For problems of this type you must correctly draw and label a figure to illustrate the given information. Then you must find equations to represent the perimeter and area. The equation for the area will be a quadratic which when graphed will be a parabola "opening down." To maximize the area, you must find the ycoordinate of the parabola's vertex. To do this we first used the vertex formula to find the xcoordinate and then substituted it back into the area formula.

D Saye


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