Extra Practice Design Problems in 3D

Extra Practice: 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. Extra Practice: Design Problems in Three Dimensions

1CHAPTER Extra Practice: Design

Problems in Three Dimensions

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

Extra Practice: Design Problems in Three Dimensions

1. Come up with at least 5 rectangles with a perimeter of 24 inches. Which rectangle has the biggest area?

2. If you did not consider a square in #1, compare the area of a square with perimeter 24 inches to the other rectangles
that you came up with. What do you notice?

3. A pentagon has a given perimeter. Make a conjecture about what type of pentagon with this perimeter will have
the biggest area.

4. A rectangular prism has a given surface area. Make a conjecture about what type of rectangular prism will have
the biggest volume.

A new peanut butter company wants to stand out compared with the competition, so decides to design a container
that is a truncated pyramid. The top of the container is a square that is 4 inches by 4 inches. The bottom of the
container is a square that is 2 inches by 2 inches. The container is 4 inches tall.

5. If the pyramid hadn’t been truncated, how tall would it have been?
6. Find the volume of the truncated pyramid.
7. Find the surface area of the truncated pyramid.
8. One cubic inch holds 0.45 ounces of peanut butter. How much peanut butter can this container hold?
9. A typical peanut butter jar is a cylinder with a height of 4 inches and a diameter of 3.9 inches. Compare and
contrast this type of jar with the new truncated pyramid container.
An open faced box is being made from a square piece of paper that measures 10 inches by 10 inches. The box will
be made by cutting small congruent x by x sized squares out of each corner.

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10. What’s an equation that relates x to the volume of the box?
11. Graph the equation from #10 and explain what it shows.
12. What size squares should you remove from each corner to maximize the volume?
13. Why does it not make sense to try to minimize the volume of the box?
14. The length of the side of the original piece of square paper is s. Come up with an equation that relates s, x, and
the volume of the box.
15. Use your equation to verify that a 15 inch by 15 inch piece of paper with 2 inch by 2 inch square corners removed
will produce a box with a volume of 242 in3.

References

1. . . CC BY-NC-SA

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www.ck12.org Chapter 1. Extra Practice: Design Problems in Three Dimensions

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