The owner of the Rancho Los Feliz has 3000 yd of fencing...

Name: Problem 3, Chapter 4: Applied Calculus for the MLSS A Brief Approach
Uploaded: 2021-03-25T15:49:25-08:00
Duration: 2 min 52 s
Channel: Sheryl Ezze
Description: The owner of the Rancho Los Feliz has 3000 yd of fencing with which to enclose a rectangular piece of grazing land along the straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area that he can enclose? What is this area?

The owner of the Rancho Los Feliz has 3000 yd of fencing with which to enclose a rectangular piece of grazing land along the straight portion of a river. If fencing is not required along the river, what are the dimensions of the largest area that he can enclose? What is this area?

Key Concepts

Optimization Under Constraints

This concept involves finding the maximum or minimum value of a function subject to certain restrictions or limitations. In many practical problems, such as maximizing the area enclosed by a fixed length of fencing, a constraint (like a fixed perimeter) reduces the number of independent variables, allowing the optimization problem to be solved using techniques from calculus or algebra.

Formulating Constraint Equations

In optimization problems, the constraints provided by the problem (for example, a fixed amount of fencing or a predetermined perimeter) are used to create an equation that relates the variables. This constraint equation is crucial as it allows one to express one variable in terms of the others, thereby reducing the dimensionality of the problem and simplifying the process of finding the maximum or minimum value.

Quadratic Function Maximization

When the objective function (such as the area to be maximized) is expressed in terms of a single variable, it often takes the form of a quadratic function. Quadratic functions have a parabolic graph, and if the coefficient of the squared term is negative, the parabola opens downward, ensuring that the vertex represents the maximum value. Finding the vertex of the parabola, either by completing the square or using calculus, yields the optimal solution.

Differential Calculus in Optimization

Calculus provides a systematic method for finding maximum and minimum values of functions. By taking the derivative of the objective function and setting it equal to zero, one can find critical points. Analyzing these points (or utilizing the second derivative test) allows one to determine whether they correspond to a maximum or minimum value. This technique is fundamental for solving continuous optimization problems.

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