TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

Prerequisites

Formation of function from description, optimization of a function of one variable.

Time allotment - time management

This problem is a somewhat standard optimization problem and with a diagram the students can formulate the geometric and economic optimization function in 10 minutes, the latter being a bit more difficult as they have to assign the various costs to specific pieces.

Expectations

We would expect the student to formulate an objective function - in the first problem one of volume and in the second problem one of profit = revenue - cost.

Further we would expect the student to be able to optimize such objective functions and compare the results.

Future payoffs

This problem demands careful reading, extraction of a drawing from a description if no drawing is provided, and a step-by-step accounting for all costs involved. Such care in accounting for terms in the objective function will be useful throughout future modeling activities.

Extensions

This IS an extension of the standard maximum volume box out of a rectangle - only this time we require a top and are concerned with economics.

One could require double thickness of the bottom, using a piece from the unused material and costing for the glueing of the two pieces.

One could discuss marginal increases in say total cost, for marginal increases in any component's cost.

References and Sources

We appreciate the suggestions and corrections offered by Paul H. Bouknecht who teaches AP Calculus at Eau Gallie High School in Melbourne FL.