# 10. The Draining Tank Problem

Students will be able to create 3D containers with an appropriate flow rate, determine how fast the water in a 3D tank is dropping, and verify the flow rate of a 3D object using differentiation.

Standards (From the California State Calculus Standards)
4.0 Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability.

9.0 Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.

12.0 Students use differentiation to solve related rate problems in a variety of pure and applied contexts.

Standards for Mathematical Practice
CCSS.MATH.PRACTICE.MP2: Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3: Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4: Model with mathematics.
CCSS.MATH.PRACTICE.MP5: Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6: Attend to precision.
CCSS.MATH.PRACTICE.MP7: Look for and make use of structure.

Learning Objectives

• Students will be able to create 3D containers with an appropriate flow rate.
• Students will be able to determine how fast the water in a 3D tank is dropping.
• Students will be able to verify the flow rate of a 3D object using differentiation.

Recommendations
Group Size: 3 to 4 students, depending on how many variations you would like students to compare within the class setting.

Class Size: up to 36 students

Materials Required

Introduction

• Students have an understanding of volume and its formulas (depending on the shape).
• students have a good understanding of 3D modeling. Prior to incorporating this lesson into a unit, it is recommended that students have had training on Google SketchUp.
• Students have a solid understanding of differentiation and integration in a Calculus setting.

To begin the lesson, review the concepts of differentiating with respect to x.  For a quick refresher about how to differentiate to find flow rate, please watch this video.  In fact, it may be a good idea to show the video to the class as a hook for the lesson.  Before getting into the meat of the lesson, it may be useful to review integration and differentiation with respect to flow rate problems.

The Meat
On paper, students sketch a container that will have a change of height of -0.5 cm per second.

Dimensions need to be designated on the drawing and appropriate formulas for volume must be included in the notebook.  For a basic representation, students will use a cylinder.  Once completed, instructor signs off on the sketch for accuracy and posts the dimensions on the board for students to see.

Using a 3D modeling program such as Google Sketchup, students create an object that represents their sketch in their notebook.  Prior to printing, measurements need to be verified by a partner, then by the instructor.  Once verified, student will send the product for printing.

During the design process, ensure that students are creating an interior and exterior dimension.  To lower the cost and time of printing, making the walls of the box less than 1 cm would be effective.

After printing, have students use a ruler to measure the dimensions and internal volume of the product and note each in their notebook.

Discussion
Using measuring cups filled with water, students will verify the interior volume of their products by filling them up and measuring the volume of water that fits.  After verification, students will time the flow of water as it is emptied as well as the change in height.

If everything has been done correctly with their calculations, the math should match up with the tests.  Regardless of the result, have students reflect on the process to see what they would improve, areas to modify, and check in with neighbors to emphasize variations.

Desired Outcomes
Due to the openness of the challenge, there are really no ideal desired outcomes, as long as the height change is -0.5 cm per second.  If this is unreasonable for your students, choose a height change challenge that will meet their level.  As the instructor, be sure to verify student work prior to designing the object in Google SketchUp.

Some Possible Extensions/Modifications
Change the objective to meet a specific need, such as:

“Students will be able to create a watering tool for a desired flow rate of 1 gallon per day.”

To modify the project, the instructor can change the challenge from “change of height” to “flow rate”.

To account for the variability of student designs, it may be helpful to have a challenge to see who can get closest to a height change of -0.5 cm per second with their printed products.

Possible Application
Most of us would love to have a self-watering plant routine.  Pour in some water and your plants stay happy.  By developing something that is supportive of this, students have an opportunity for entrepreneurship.  Have students look up the desired water intake for a fruit tree, vegetable plant, or shrub.  Once they have it, see if they can design and print a tank that meets the needs by filling it with water once per day (beginner) or week (advanced), if even possible Airwolf 3D’s printable dimensions.

Questions to Ponder

• What were the variables that were involved?
• What were the constants?
• What would happen if you changed (choose a variable)?
• What would happen if the dimensions were changed in the tank?

Content & Instruction Developed by:
John Stevens – Airwolf 3D STEM Consultant
Instructional Coach – Technology
Chaffey Joint Union High School District
CUE Rockstar Faculty & Organizer