This activity works well in a multivariable calculus class, after techniques for solving constrained optimization problems have been discussed.  

Pre-Class Assignment

Students begin by completing a pre-class reading assignment pertaining to fish farm canal design accompanied by a short quiz.  This gives them the background needed to tackle the in-class portion of the assignment. 

In-Class Work

Students should work in groups of 2 or 3 to complete the in-class portion of the assignment. Each student should be given a copy of the activity handout.  Responses should ultimately be recorded on the handout, but having access to whiteboard space may help students communicate their ideas with each other.  At least one student in each group should have a computer with Mathematica or other software capable of simultaneous equation solving installed.  The paragraphs below briefly describe the four parts of the handout.

Part 1 - Define Your Objective Function

Here students establish that the objective is to minimize the wetted perimeter in order to maximize the flow rate through a trapezoidal canal having a prescribed cross-sectional area. Students then use a combination of geometry and trigonometry to define the objective function for the optimization.

Part 2 - Develop Your Constraint Equation

In this section of the assignment, students again apply their geometry and trigonometry knowledge to write an equation that represents the wetted cross-sectional area of a trapezoidal canal in terms of the height, diameter, and side slope. This equation becomes a constraint equation when the wetted cross-sectional area is fixed at 1.5 square meters, the design specification given in the problem statement. 

Part 3 - Optimize

In this section of the assignment, students determine the canal height, diameter, and side slope needed to maximize the flow rate through a trapezoidal canal with a wetted cross-sectional area of 1.5 square meters. Some students will opt to use the constraint equation to reduce the number of variables in the objective function from 3 to 2, and then identify the inputs to the objective function that minimize the wetted perimeter. Others will use the Lagrange multiplier method on objective function and constraint equation produced in Parts 1 and 2. Both approaches are outlined in the key attached to this card. To save time, I recommend that students use Mathematica or a similar computational tool to solve the system of equations they set up in Part 3.

Part 4 - Review and Reflect

Either in class or after class, students are prompted to answer a brief set of questions that make it clear why a trapezoidal canal shape is preferred to a rectangular one in order to maximize flow rate.