Worked Examples Design Problems in 3D

Worked Examples: Design
Problems in Three Dimensions

CK-12
Kaitlyn Spong

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Kaitlyn Spong

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Printed: July 14, 2017

www.ck12.org Chapter 1. Worked Examples: Design Problems in Three Dimensions

1CHAPTER Worked Examples: Design

Problems in Three Dimensions

Learning Objectives
Here you will apply geometric methods to solve design problems involving optimization in three dimensions.

Worked Examples: Design Problems in Three Dimensions

Example 1

An ice cream cone is made from a thin wafer cookie that is rolled to make the cone shape. The radius of the ice
cream cone is 2.5 centimeters and the slant height is 11.5 centimeters. How many square centimeters of wafer cookie
are needed to make each ice cream cone?

Solution:

When unwrapped, the cone is a sector of a circle. The arc length is the circumference of the base of the cone (2πr =
2π(2.5) = 5π cm) and the radius is the slant height (11.5 cm).

To find what fraction of the circle this is, note that the circumference of the full circle would have been 2π(11.5) =

23π. Therefore, this is 5π = 5 of the circle. To find the area needed for the ice cream cone, find the area of the
23π 23

sector.

A= 5 πr2 = 5 π(11.5)2 = 90.32 cm2
23 23

Use the information from the problem above for these questions.

Example 2

To save money, you decide you could either make the cones skinnier or shorter. Which change will cause the biggest
change in the surface area: shortening the radius to 2 centimeters or shortening the slant height to 11 centimeters?
Solution:
If the radius is 2 cm, the new surface area will be 72.26 cm2. If the slant height is 11 cm, the new surface area will
be 86.39 cm2. The biggest change in surface area is caused by shortening the radius.

Example 3

Waffle cones have a radius of 4 centimeters and a slant height of 16 centimeters. How much bigger is the surface
area of the waffle cone compared to the surface area of the regular ice cream cone?
Solution:
The surface area of the waffle cone is 201.06 cm2. This surface area is 110.74 square centimeters bigger than a
regular ice cream cone.

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Example 4

Come up with a function that outputs the area of wafer cookie needed per cone given the radius and slant height of
the cone.

Solution:

Let l = slant height and r = radius. A = πl2 · 2πr = πrl.
2πl

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www.ck12.org Chapter 1. Worked Examples: Design Problems in Three Dimensions

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