““The Scoop” on The Scoop” on the Expanding Ice the Expanding Ice

Cream ConeCream ConeTimothy Wong

Lexi Trumble

Once Upon a Time…There was a small child named Bruster. Our dear There was a small child named Bruster. Our dear

friend Bruster was great companions with his two friend Bruster was great companions with his two neighbors, Ben and Jerry. On one early Saturday neighbors, Ben and Jerry. On one early Saturday morning, while their parents were still asleep, the morning, while their parents were still asleep, the three boys heard the magical sound of the ice three boys heard the magical sound of the ice cream truck driving on their street. They jumped cream truck driving on their street. They jumped out of bed and ran to the end of their driveway, out of bed and ran to the end of their driveway, eagerly anticipating the delectable treat they eagerly anticipating the delectable treat they would soon receive. They met the ice cream man would soon receive. They met the ice cream man with wide smiles and money in hand. They with wide smiles and money in hand. They selected their ice cream treat with care and selected their ice cream treat with care and gleefully unwrapped the source of happiness. The gleefully unwrapped the source of happiness. The ice cream man drove away and the boys sat in ice cream man drove away and the boys sat in their driveways to enjoy their sugar rush. Soon, their driveways to enjoy their sugar rush. Soon, however…however…

…they realized with a start that their ice cream cones were expanding! While some children would be filled with delight at this prospect, the children at hand recognized the peril of their situation. They resorted to the mathematical concepts in their brains to realize that this was, indeed, a real-world application of “Rate of Change” calculus!!!

The volume The volume VV of a cone is increasing at the rate of a cone is increasing at the rate of 28π cubic units per second. At the instant of 28π cubic units per second. At the instant when the radius when the radius rr of the cone is 3 units, its of the cone is 3 units, its volume is 12π cubic units and the radius is volume is 12π cubic units and the radius is increasing at ½ unit per second.increasing at ½ unit per second.

At the instant when the radius of the cone is 3 units, what is the rate of change of the area of its base?

b) At the instant when the radius of the cone is 3 units, what is the rate

of change of its height h?

c) At the instant when the radius of the cone is 3 units, what is the instantaneous rate of the change of the area of its base

with respect to its height h?