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# 2. Students Use 3D Printing to Verify the Flow Rate of a 3D Object Using Differentiation

## The Draining Tank Problem

### Standard (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:

Paper and pencil for drafting
Cups, vases, or other 3D models to show students relevance of flow rate (must be destructible – not your mother’s favorite vase!)
Airwolf 3D Printer
Water
Measuring Cup (for verification)
Stopwatch
Ruler to verify dimensions

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For a video on creating a 3D model, go to http://youtu.be/gsfH_cyXa1o
For a video on creating a How to create a Cylinder with a Spigot, go to http://youtu.be/5o3UWBoNDZQ

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

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