Fencing a Horse Corral Carol has 2400 ft of fencing to fence in a rectang horse corral. a. Find

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Fencing a Horse Corral Carol has 2400 ft of fencing to fence...

Fencing a Horse Corral Carol has $2400 \mathrm{ft}$ of fencing to fence in a rectang horse corral.
a. Find a function that models the area of the corral in terms of the width $x$ of the corral.
b. Find the dimensions of the rectangle that maximize the area of the corral.

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Key Concepts

Quadratic Functions

Quadratic functions, which typically have the form ax² + bx + c, arise in many optimization problems. When the area is expressed as a quadratic function of one variable, its graph is a parabola. The vertex of this parabola represents the maximum (or minimum) value, which in this case gives the dimensions that maximize the area.

Function Modeling

Function modeling is the process of representing a real-world scenario with a mathematical function. For a rectangle with a fixed perimeter, one dimension is expressed in terms of the other using the perimeter constraint, and then this relationship is substituted into the area formula to create a function of a single variable.

Perimeter Constraint

This concept refers to the fixed amount of boundary (fencing) available to form a figure. It imposes a linear relationship between the dimensions of the rectangle, ensuring that the sum of its sides equates to the given total perimeter.

Optimization

Optimization involves finding the best solution, such as a maximum or minimum, under specific constraints. In the context of geometry, this often means determining the dimensions of a shape that yield the maximum area or minimum perimeter, given a fixed resource.

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