Rate of Change - Bubblegum

We're used to seeing formulas that tell us, for example, that the volume of a sphere is (4/3)π times the cube of the radius. However, we don't often think about how the volume changes as the radius changes. Think about blowing a gum bubble, or filling a balloon from a tank of helium. If the helium is dispensed at a constant rate (meaning that the rate of change of the balloon's volume is constant), we observe that the balloon grows in diameter quickly at first, but more slowly as time goes on. It takes more gas to add an inch of diameter when the balloon is large than it does to add an inch of diameter when the balloon is small. By the same token, in order for the balloon's diameter to grow steadily, the volume would have to grow at a faster rate (the flow rate of the helium would continully increase) as the balloon grows.
Use the Switch button to select one of the two cases, either Constant Airflow or Constant Change in Radius. Click Reset to clear any existing traces and to set the gum bubble's size to zero. Then click Start/Stop to observe how the bubble grows. Plots of the radius and volume are traced as the bubble grows, to illustrate the "shapes" of these functions over time.