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Balloon Inflation Simulator

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About This MicroSim

This interactive simulation helps students discover why the rate of radius change (dr/dt) decreases as a balloon inflates, even when air flows in at a constant rate. The key insight is the inverse square relationship between dr/dt and the balloon's radius.

What You'll See

  • An animated expanding balloon that grows as air flows in
  • Volume flow visualization showing air entering the balloon
  • Three synchronized graphs showing:
    • V(t): Volume increasing over time
    • r(t): Radius increasing over time (but with decreasing slope)
    • dr/dt: Rate of radius change decreasing over time
  • Current values panel displaying V, r, dV/dt, dr/dt, and surface area

The Key Formula

\[\frac{dr}{dt} = \frac{dV/dt}{4\pi r^2}\]

This formula reveals why the balloon radius grows more slowly as the balloon gets bigger:

  • The volume flow rate (dV/dt) stays constant (you're blowing air in at the same rate)
  • But the surface area (4 pi r squared) keeps increasing
  • So the same amount of air has to spread over more surface area
  • Result: the radius increases more slowly as the balloon grows

Delta Moment

"Watch what happens as I inflate this balloon! At first, a little puff of air makes a big difference. But as the balloon gets bigger, that same puff barely moves the surface. It's like trying to stretch a bigger and bigger rubber band with the same force!"

Embedding This MicroSim

You can include this MicroSim on your website using the following iframe:

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<iframe src="https://dmccreary.github.io/calculus/sims/balloon-inflation-simulator/main.html"
        height="640px"
        width="100%"
        scrolling="no"
        style="border: none;">
</iframe>

Lesson Plan

Learning Objective

Students will compare how the rate of radius change varies with different radii and volume flow rates, discovering the inverse relationship.

Bloom's Taxonomy Level: Analyze (L4) Bloom's Verbs: Compare, Analyze, Discover

Grade Level

High School (Grades 11-12) - AP Calculus AB/BC

Duration

15-20 minutes

Prerequisites

  • Understanding of related rates concepts
  • Familiarity with the chain rule
  • Volume formula for a sphere: V = (4/3) pi r cubed
  • Surface area formula: A = 4 pi r squared

Warm-Up Questions

Before starting the simulation, ask students:

  1. If you blow air into a balloon at a constant rate, does the radius increase at a constant rate?
  2. Why might the rate of radius change depend on the current size of the balloon?
  3. What happens to the surface area as the balloon grows?

Activities

Activity 1: Initial Exploration (5 minutes)

  1. Click Play and watch the balloon inflate
  2. Focus on the dr/dt graph (purple) - what shape does it have?
  3. Compare to the r(t) graph (blue) - how does the slope change?
  4. Discuss: Why does dr/dt decrease even though dV/dt is constant?

Activity 2: Varying Flow Rate (5 minutes)

  1. Reset the simulation
  2. Set dV/dt to 50 cm cubed per second and run
  3. Reset and set dV/dt to 150 cm cubed per second
  4. Compare: Does the dr/dt graph have the same shape? Does it start higher or lower?
  5. Conclusion: Higher flow rate means faster initial growth, but the decrease pattern is the same

Activity 3: Mathematical Analysis (5 minutes)

Starting from V = (4/3) pi r cubed, derive the relationship:

  1. Differentiate both sides with respect to time t
  2. dV/dt = 4 pi r squared times dr/dt (using chain rule)
  3. Solve for dr/dt: dr/dt = (dV/dt) / (4 pi r squared)
  4. Verify: This explains why dr/dt decreases as r increases (inverse square)

Activity 4: Connection to Surface Area (5 minutes)

  1. Notice that 4 pi r squared is the surface area formula
  2. Interpretation: The rate of radius change depends on how much surface area the new air must "stretch"
  3. Physical intuition: Same air flow, bigger surface means smaller thickness increase

Discussion Questions

  1. At what point is dr/dt largest? (When r is smallest, at the start)
  2. Does dr/dt ever reach zero while inflating? (No, but it approaches zero as r grows)
  3. If you doubled the flow rate, how would the dr/dt graph change? (It would be exactly twice as tall at every point)
  4. Real balloons eventually pop. What physical factor does this simulation ignore? (Elastic limit, pressure changes)

Assessment

Quick Check: If a balloon has radius 5 cm and air flows in at 100 cm cubed per second, calculate dr/dt.

Solution: dr/dt = 100 / (4 pi times 25) = 100 / (100 pi) approximately equals 0.318 cm/s

Extension Problem: Two identical balloons start at different sizes (r = 2 cm and r = 4 cm). If air flows into both at the same rate, which balloon's radius grows faster? By what factor?

Solution: The smaller balloon grows faster. The ratio of rates is (4 squared) / (2 squared) = 4. The small balloon's radius increases 4 times faster!

This simulation connects directly to classic related rates problems:

  • Given: dV/dt (constant rate of volume increase)
  • Find: dr/dt (rate of radius change)
  • Key insight: dr/dt depends on current r, creating a non-constant rate of change

This is why students must set up the equation first, then substitute values - unlike simpler problems where rates are constant.

References