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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.
An expression for this figure's perimeter would be:
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:

square yards
Shown below is a graph of our area function .
Therefore the largest area that the farmer could enclose would be a square where each side has a length 250 yards.


Let's Practice

Question #1
AudioThe owner of a ranch decides to enclose a rectangular region with 140 feet of fencing. To help the fencing cover more land, he plans to use one side of his barn as part of the enclosed region. What is the maximum area the rancher can enclose?

Question #2
AudioA farmer wishes to enclose a rectangular region bordering a river using 600 feet of fencing. He wants to divide the region into two equal parts using some of the fence material. What is the maximum area that can be enclosed with the fencing?

Try These
Question #1
A rancher has 1200 feet of fencing to enclose two adjacent rectangular corrals of equal lengths and widths as shown in the figure below. What is the maximum area that can be enclosed in the fencing?

Question #2
AudioA local grocery store has plans to construct a rectangular parking lot on land that is bordered on one side by a highway. There are 1280 feet of fencing available to enclose the other three sides. [Let x represent the length of the two parallel sides of fencing.] Find the dimensions that will maximize the area of the parking lot.

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 y-coordinate of the parabola's vertex. To do this we first used the vertex formula to find the x-coordinate and then substituted it back into the area formula.

D Saye

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