Maximize the Volume of a Box: Exploring Optimization Using Calculus

The process of finding the conditions for a function that represents a certain amount of required effort or which represents a certain needed benefit that gives minimum and maximum values is called optimization. Beginning calculus students are often presented with this type of optimization problem:

We want to make a box with a square base and we only have 12 square meters of cardboard to use in construction of the box. If all of the cardboard is used, what is the maximum volume that the box can have?

This type of optimization problem is subject to constraint. Constraint is a condition that is true no matter what. We will find either the largest or smallest value of a function. The constraint usually is described by an equation: in this case, write down what you know after you carefully read the problem. Second, identify the quantity you need to optimize, and the condition, or constraint.

In the problem noted above, one quantity, 12 square meters is clearly identified as it is the amount of material used, so that is your constraint as it is a fixed value. Think of it also as the surface area of the box. Sketch it out. Use your maximization and constraint equations and solve for h. Plug the value of h into the equation for volume function and you will end up with the first and second derivatives. We know that the width is greater than zero in this problem, so setting the first derivative to zero will garner two critical points. The positive answer is, of course, the length of the box, although some equations have valid negative answers. To optimize the volume of this box, we only need the maximum volume.

What is Calculus?: Things change all the time. Calculus provides a way to study that change and to deduce or predict consequence of that change.

Volume and Surface Area: Lateral and surface areas, Cavalieri’s Principle, and volume formulas as relating to prisms, cylinders, pyramids, cones, spheres.

Optimization Examples: This tutorial explores method of applying derivatives in order to calculate simple maximization using a fixed quantity and a constraint. This tutorial is part of an e-book for beginning calculus.

Absolute Extrema: This very useful video explains the concept of absolute extrema and optimization.

Global Extrema: Lab-style exercise in locating global extrema, with sample problems to illustrate the term.

Interactive Optimization: Use the slider applet in order to solve the example problems.

Linear Optimization: The structured process includes an overview of the optimization modeling process and applications.

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