The purpose of this optional culminating lesson is to give students more experience working with nonlinear functions that arise out of the work students have been doing with the volumes of cylinders, cones, and spheres.

The Warm-up invites students to contrast how knowing the height of a cylinder does not fix the volume or radius of the cylinder in the same way it does for a sphere of fixed height. Next, students continue to work with spheres and reason about how scaling the radius affects the volume of a sphere. First they repeat calculations for the volume of spheres of various radii, and then they generalize for spheres with radius (MP8).

In the last activity, students work with three different functions (represented three different ways) showing how the height of water in three different shapes is a function of the volume of water being poured into the shape. They consider questions such as:

• Which container is largest?
• At what volume of water poured is the height of water in the containers the same?
• For what range of water volume poured does a particular container have the greatest height of water?
• How do the representations show the maximum height of each container?