You have 80 yards of fencing to enclose a rectangular region. Find the dimensions of the rectang

step 1 So let's consider the rectangular region here and we define the dimension as x and the y here. A

You have 80 yards of fencing to enclose a rectangular...

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Key Concepts

Perimeter Constraint

This key concept involves using the known total boundary length to form an equation that relates the dimensions of a geometric shape. For a rectangle, the perimeter is expressed as 2L + 2W, and setting this equal to a fixed number allows one to express one variable in terms of the other, thereby reducing the number of independent variables in the problem.

Quadratic Function Optimization

When the area of the rectangle is expressed in one variable using the perimeter constraint, the resulting equation is a quadratic function. Quadratic functions typically have a parabolic shape, and if the parabola opens downward, it has a maximum point at its vertex. Finding this vertex yields the dimensions that maximize the area.

Maximization Strategy

Optimization involves strategies such as using derivatives or applying the vertex formula to determine the maximum value of a function. In problems where constraints are present, such as a fixed perimeter, these strategies help in systematically identifying the optimum dimensions that achieve the best possible outcome, like the maximum enclosed area.

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