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Can Optimizer

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Description

This MicroSim demonstrates one of the classic optimization problems in calculus: finding the dimensions of a cylindrical can that minimize the surface area for a given volume. Students explore how changing the radius affects the height (since volume is fixed) and consequently the total surface area.

The MicroSim features:

  • 3D Cylinder Visualization: A rotating 3D cylinder showing the current dimensions with color-coded surface areas (blue for top/bottom circles, orange for the lateral surface)
  • Surface Area Graph: A real-time plot of S(r) showing how surface area varies with radius, with the optimal point marked in green
  • Interactive Controls: A slider to adjust radius and an input field to change the volume
  • Data Panel: Shows all calculated values including the percentage above the minimum surface area
  • Compare Mode: Displays three cans (tall/thin, optimal, short/wide) to visualize why the optimal dimensions work

The Math Behind It

For a cylinder with volume V and radius r:

  • Height: h = V / (pi r^2)
  • Surface Area: S(r) = 2 pi r^2 + 2 pi r h = 2 pi r^2 + 2V/r

To minimize, we take the derivative and set it equal to zero:

  • dS/dr = 4 pi r - 2V/r^2 = 0
  • Solving: r = (V / (2 pi))^(1/3)

At this optimal radius, the height equals the diameter (h = 2r), creating a "square" profile when viewed from the side.

Lesson Plan

Learning Objectives

By the end of this activity, students will be able to:

  1. Calculate optimal cylinder dimensions for a given volume constraint
  2. Explain why the optimal can has height equal to diameter
  3. Apply derivative techniques to solve real-world optimization problems
  4. Analyze how changing constraints affects optimal solutions

Grade Level

High School (Grades 11-12) and AP Calculus

Prerequisites

  • Understanding of derivatives
  • Basic knowledge of cylinder geometry (volume and surface area formulas)
  • Familiarity with setting derivatives equal to zero to find extrema

Duration

20-30 minutes

Activity Sequence

Part 1: Exploration (5-7 minutes)

  1. Start with default values (V = 1000 cm^3)
  2. Move the radius slider from minimum to maximum
  3. Observe how the cylinder shape changes and how the surface area responds
  4. Find the approximate optimal radius by watching the graph

Part 2: Mathematical Analysis (8-10 minutes)

  1. Click "Find Optimal" to jump to the optimal radius
  2. Record the optimal radius and height
  3. Calculate h/2r to verify h = 2r at optimum
  4. Derive the formula by hand and verify it matches

Part 3: Comparative Analysis (5-8 minutes)

  1. Enable "Compare" mode to see three cans side by side
  2. Discuss why extremes (tall/thin or short/wide) waste material
  3. Change the volume to 500 cm^3 and observe how optimal dimensions scale
  4. Notice that the shape (ratio h:r) stays the same regardless of volume

Part 4: Extension Questions

  1. What if we only need to minimize the lateral surface (open-top can)?
  2. How would the answer change if material costs differed for top/bottom vs. sides?
  3. Why don't real soup cans exactly match the mathematical optimum?

Assessment Ideas

  • Have students predict the optimal radius for a new volume before using the simulator
  • Ask students to explain in writing why the "square profile" minimizes surface area
  • Calculate the material savings (percentage) compared to a poorly designed can

Connection to AP Calculus

This activity directly addresses:

  • Applications of Derivatives: Using derivatives to solve optimization problems
  • Critical Points: Finding where derivative equals zero
  • Second Derivative Test: Verifying minimum (though not shown in MicroSim, can be done analytically)
  • Mathematical Modeling: Translating a real-world problem into calculus

References