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Week 10: Geometry - Circles


Maximal Volume of a Cone (Printable PDF)

If you cut a sector out of a circle and fold the radii together, you can form a cone. What central angle of the sector will produce a cone with maximum possible volume?

 

GPS

As seen in Problem Exploration

M6M3. Students will determine the volume of fundamental solid figures (right rectangular prisms, cylinders, pyramids, and cones).

b. Compute the volume of fundamental solid figures, using appropriate units of measure.

c. Estimate the volumes of simple geometric solids.

d. Solve application problems involving the volume of fundamental solid figures.

For each central angle, the students determine the volume of the cones that it creates by understanding how to use the formula for the volume of a cone (i.e. what each variable means). 

The students can begin to estimate the different volumes of cones that are created as the central angle changes. 

M6G2. Students will further develop their understanding of solid figures.

b. Compare and contrast cylinders and cones.
Students can discuss differences in cylinders and cones in terms of creating them and finding their volumes
M6A2.  Students will consider relationships between varying quantities.

a. Analyze and describe patterns arising from mathematical rules, tables, and graphs
 
Students can create a table with the different values of the central angle and the volumes of the made cones.

By plotting the data, they can determine a potential maximum volume.

M7A3. Students will understand relationships between two variables.

b. Represent, describe, and analyze relations from tables, graphs, and formulas.
 

A table can be created that compares the central angle of the circle with the volume of the cone it creates. The data can be analyzed from the table or a graph can be drawn.

The formula for the volume of a cone can be written in terms of the central angle by knowing that the length of the arc of the circle is equal to the circumference of the base of the cone.

M8A3. Students will understand relations and linear functions.

i. Identify relations and functions as linear or nonlinear.

j. Translate among verbal, tabular, graphic, and algebraic representations of functions.

In graphing the relationship between the central angle and the volume of the cone it creates, students should recognize that it is a nonlinear relationship.