Solve each problem. Area of a Rectangular Region A farmer...

Solve each problem. Area of a Rectangular Region A farmer wishes to enclose a rectangular region bordering a barn with fencing, as shown in the diagram. Suppose that $x$ represents the length of each of the three parallel pieces of fencing. She has 600 feet of fencing available. (a) What is the length of the remaining piece of fencing in terms of $x ?$ (b) Determine a function $\mathscr{A}$ that represents the total area of the enclosed region. Give any restrictions on $x$ (c) What dimensions for the total enclosed region would give an area of $22,500$ square feet? Determine the answer analytically. (d) Use a graph to find the maximum area that can be enclosed. (IMAGE CAN'T COPY)

Solve each problem.
Area of a Rectangular Region A farmer wishes to enclose a rectangular region bordering a barn with fencing, as shown in the diagram. Suppose that $x$ represents the length of each of the three parallel pieces of fencing. She has 600 feet of fencing available.
(a) What is the length of the remaining piece of fencing in terms of $x ?$
(b) Determine a function $\mathscr{A}$ that represents the total area of the enclosed region. Give any restrictions on $x$
(c) What dimensions for the total enclosed region would give an area of $22,500$ square feet? Determine the answer analytically.
(d) Use a graph to find the maximum area that can be enclosed.
(IMAGE CAN'T COPY)

Key Concepts

Linear Relationships

In many problems involving dimensions and constraints, a linear relationship is used to relate different quantities. In this example, the total amount of material available (fencing) is distributed among the sides of a rectangle, leading to an expression where the length of one side is defined as a linear function of another variable that represents the other sides. This is central to setting up the problem correctly.

Quadratic Functions

When one dimension of a rectangle is expressed in terms of another and the area is calculated as the product of these expressions, a quadratic function naturally arises. Quadratic functions are essential in modeling the area as a function of one variable and are pivotal for further analysis such as finding maximum or minimum values.

Optimization

Optimization in the context of this problem refers to finding the maximum or minimum value of the area function. With the quadratic function obtained from the area of the rectangle, techniques such as completing the square or using calculus are used to determine the dimensions that yield the maximum area, which is a key concept in optimization problems.

Constraints and Domain Restrictions

Physical constraints, such as having a positive length for each side of a rectangle, impose domain restrictions on the variable used. These restrictions ensure that the solutions are meaningful in a real-world context, and they are necessary to correctly interpret the mathematical model.

Graphical Analysis

Graphical analysis involves plotting the quadratic area function to visualize its behavior, including the vertex which represents the maximum area. It aids in verifying the analytical solution and provides insight into how changes in the variable affect the overall area, making it a useful tool in comparing theoretical and actual outcomes.

Solving Quadratic Equations

The process of solving quadratic equations, such as using the quadratic formula, is a crucial step when determining specific dimensions that satisfy a given area. It allows for the analytical determination of dimensions by finding the roots of the quadratic equation that arises from setting the area function equal to a specified value.

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