You have learned how to apply the chain rule and to solve problems involving rates of change In this task you will solve a problem involving a combination of these ideas Hypothesis The rate of change of the volume of a sphere is proportional to its surface area Experiment a Obtain a spherical lollipop on a stick Assume that it is a perfect sphere Measure and record the initial radius of this sphere Hint Remember the relationship between radius and circumference b Place the lollipop in your mouth and carefully consume it uniformly for 30 s Measure and record the new radius in a table similar to the one here Repeat until you have at least ten measurements Time s Radius of Lollipop cm 0 30 60 90 120 150 180 210 240 270 300 c Use your data to determine the rate of change of radius with respect to time Write an equation to model the radius as a function of time Justify your choice of models d Write the formula to model the volume as a function of the radius Use your equation from part c to model the volume as a function of time e Use your model from part c to calculate the rate of change of volume with respect to time when the radius has reached half its original value f Explain why you should not expect the rate of change of volume with respect to time to be constant in this situation g Use your model to estimate how much time is required to completely consume the lollipop h Does this experiment confirm the hypothesis Explain i How would the time to consume the lollipop change if the initial radius were multiplied by a constant k Justify your response