# Activities

• • • ##### Subject Area

• Math: Geometry: Surface Area and Volume

• ##### Author College

60 Minutes

• ##### Device
• TI-89 / TI-89 Titanium
• TI-92 Plus / Voyage™ 200

## Optimisation

#### Activity Overview

This activity will illustrate how geometric problems like finding the minimum/maximum area and volume of a geometric figure can be calculated using the Computer Algebra System CAS tool.

#### Before the Activity

• See the attached PDF file for detailed instructions for this activity
• Print pages 1 - 2 from the attached PDF file for the class
• #### During the Activity

Distribute the pages to the class.

• Write the formula for computing the volume of the cylinder in terms of height and radius of base
• Solve this equation manually in terms of height
• Write the formula for finding the surface area of the cylinder
• Substitute the value of the height [in terms of radius] into the formula for area
• Study the Area formula in terms of radius and find its derivative
• Equate the derivative to zero and solve for the radius
• Calculate the corresponding value of height and compare the numerical values of radius and height
• Form a conjecture and prove it
• Repeat the procedure using CAS and understand the advantage of using the tool
• Identify that the minimum area is obtained when the height is equal to twice the radius

• Cut a circular piece of paper and fold it to make a cone
• Determine the height and radius of the cone in terms of the fraction of the paper and the radius of the paper
• Find the volume in terms of the fraction of the paper and the radius of the paper
• Determine the derivative and equate it to zero
• Calculate the fraction of the paper and the corresponding central angle, which give the maximum volume
• Repeat the procedure using CAS and realize that CAS is superior
• #### After the Activity

Students complete the exercises listed on the activity pages.

Review student results.

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary