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Box Volume Optimizer

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About This MicroSim

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The Box Optimizer is a classic calculus optimization problem brought to life. When you cut equal squares from the corners of a flat cardboard sheet and fold up the sides, you create an open-top box. But what size should those cuts be to maximize the volume?

This MicroSim lets you explore this question visually. You'll see:

  1. Flat Cardboard Template (left): Watch the cut squares change size as you adjust x
  2. 3D Box View (center): See the resulting box in 3D with folding animation
  3. Volume Graph (right): The function V(x) = x(L-2x)(W-2x) with your current position marked

How to Use

  1. Drag the slider to change the cut size (x) and watch all three views update
  2. Click "Animate Fold" to see the flat cardboard fold into a 3D box
  3. Click "Show Optimal" to jump directly to the maximum volume
  4. Change cardboard dimensions by clicking the input boxes and typing new values (press Enter to apply)
  5. Watch the data panel to see current dimensions, volume, and percentage of maximum

The Math Behind It

For a cardboard sheet of length L and width W, cutting squares of size x from each corner gives:

  • Box length: L - 2x
  • Box width: W - 2x
  • Box height: x

The volume function is:

\[V(x) = x(L - 2x)(W - 2x)\]

To find the maximum, we take the derivative and set it to zero:

\[V'(x) = 12x^2 - 4(L + W)x + LW = 0\]

Using the quadratic formula gives us the optimal cut size.

Delta Moment

"See how the volume graph has that beautiful peak? That's where the magic happens! Too small a cut and you get a shallow tray. Too big and you're folding up tiny walls. The sweet spot is where V'(x) = 0. Calculus helps us find it exactly!"

Lesson Plan

Learning Objectives

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

  1. Explain how changing the cut size affects each dimension of the resulting box
  2. Connect the visual representation to the algebraic volume function V(x) = x(L-2x)(W-2x)
  3. Analyze the volume function graph to identify critical points and maximum volume
  4. Apply the optimization process to find the ideal cut size for different cardboard dimensions

Suggested Activities

Activity 1: Exploration (10 min)

  • Start with default 18" x 12" cardboard
  • Have students predict: "What cut size do you think maximizes volume?"
  • Let them explore with the slider, then click "Show Optimal" to check

Activity 2: Pattern Finding (15 min)

  • Try different cardboard dimensions (square vs. rectangular)
  • Record the optimal x for each dimension set
  • Look for patterns: Is there a relationship between dimensions and optimal cut?

Activity 3: Calculus Connection (10 min)

  • Show that V'(x) = 0 gives the same answer as "Show Optimal"
  • Discuss why the derivative equals zero at maximum
  • Connect second derivative test to confirm maximum vs. minimum

Assessment Questions

  1. For an 18" x 12" sheet, approximately what fraction of the width is the optimal cut?
  2. What happens to the volume if you cut exactly W/2 from each corner? Why?
  3. If you double both L and W, what happens to the maximum volume?

References