Maximum Area Shondra wants to enclose a rectangular garden with 200 yards of fencing. What dimen

step 1 We want to find the maximum area of a rectangle of a garden when we're given 200 yards of fencin

Maximum Area Shondra wants to enclose a rectangular garden...

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Key Concepts

Optimization

Optimization involves finding the highest or lowest value of a function within a defined set of conditions. In this type of problem, the goal is to maximize an objective—in this case, the area of a rectangle—while working within specific limits such as a fixed amount of material.

Constrained Optimization

Constrained optimization refers to optimization problems that include restrictions or conditions (constraints). For example, the total length of the fence limits the combinations of the rectangle's length and width, and this constraint is used to express one variable in terms of the other for maximization.

Quadratic Functions

When the area of the rectangle is expressed as a function of one variable after applying the constraint, the function typically takes the form of a quadratic equation. Quadratic functions have a parabolic graph, and finding the vertex of this parabola reveals the optimal dimension that maximizes the area.

Vertex of a Parabola

The vertex of a parabola is the highest or lowest point of a quadratic function. In maximization problems, identifying the vertex allows one to determine the input value that results in the maximum output. This can be done by using methods such as completing the square or the vertex formula.

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