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.
Calculus Glossary: Before working in calculus, it's best to know the lingo.
What is a Derivative?: A derivative is simply a way of measuring change, denoted by the Greek symbol delta, ?. In other words, a derivative is the function of a point x and we want to know how that function is changing at that point. It is "the limit approached by the rate of change in the function when ?x becomes arbitrarily small."
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.
Free Calculus Courses Online: MIT's Open CourseWare Project has in-depth courses on optimization. Pick which one you want and open the lecture notes to view the .pdf files.
Optimization Applet: Very cool tool to find and visualize the optimization of a box. Input your information and see the shape of the box and the graphs.
Optimal Control: Overview of the calculus of variations and optimizations.
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.
The First Derivative: Tutorial on maxima and minima with formulas and reasoning.
Coursenotes and Handouts: Peruse the 24 calculus lectures that include course notes and handouts to optimize learning.
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.
Real Life Problems: These optimization examples ask "where can we reach from point A with X amount of fuel" and "what obstacles do we need to go around or take into account?"