You have 200 feet of fencing to enclose a rectangular plot...

Name: Problem 66, Chapter 3: Algebra and Trigonometry
Uploaded: 2020-06-10T12:14:16-08:00
Duration: 5 min 13 s
Channel: Suman Saurav Thakur
Description: You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

You have 200 feet of fencing to enclose a rectangular plot that borders on a river. If you do not fence the side along the river, find the length and width of the plot that will maximize the area. What is the largest area that can be enclosed?

Key Concepts

Constrained Optimization

Constrained optimization involves finding the maximum or minimum of a function subject to specific restrictions or constraints. This is a common scenario in many applied problems where resources or conditions limit the possible solutions, and it typically requires forming an equation for the objective to be optimized while considering the given constraint.

Objective Function

The objective function represents the quantity that needs to be optimized, such as maximizing area or minimizing cost. In optimization problems, this function is expressed in terms of relevant variables, and the goal is to determine the variable values that yield the optimum value of the function.

Constraint Equation

A constraint equation defines the limits imposed on the variables of the objective function, often reflecting resource limitations, physical boundaries, or other requirements. This equation is used to relate the variables to each other so that the optimization can be carried out in a reduced number of dimensions.

Quadratic Function Optimization

Many optimization problems result in a quadratic function once the constraint has been applied, especially when dealing with rectangular areas or similar shapes. Quadratic functions have a parabolic graph with a clear vertex, and the maximum or minimum value of the function can be found at this vertex using methods from calculus or by completing the square.

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