Maximizing Volume

1 / 12

# Maximizing Volume - PowerPoint PPT Presentation

Maximizing Volume. March 23, 2004. Profit and Product Distribution . . . Why does cereal come in a rectangular prism and potato chips come in a bag?. Manufacturing and Advertising . . . Why does ice cream come in so many different containers?

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## Maximizing Volume

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### Maximizing Volume

March 23, 2004

Profit and Product Distribution . . .

Why does cereal come in a rectangular prism and potato chips come in a bag?

Manufacturing and Advertising . . .
• Why does ice cream come in so many different containers?
• There are so many things that go into packaging development, one of which is trying to use the least material (which is cheaper) that will hold the most of their product while maximizing the surface area to advertise the product. Thus they are increasing profit while decreasing cost.
Maximizing Volume
• Suppose cardboard for your product comes in 8.5 x 11 sheets. You want to construct a box (ignore the top) that will hold the most of your product.
• To construct a rectangular prism from the sheet, you cut out congruent squares from the corners, then fold up the sides.
• What size corners should you cut out to get the biggest box possible, the one with the most volume?

x = 1

x = 1

Let x = 1 in.

Cut 1 by 1 in squares from each corner.

Fold up the sides to form the open box.

What are the dimensions of the box?

What is its volume?

Volume of the box (prism)?

V = AH

What are the dimensions of the base and its Area?

Length of the base =

(8.5 – 1 – 1) = 6.5

Width of the base =

(11 – 1 - 1) = 9

What is the Height of the box?

1

What is the volume?

V = (6.5)(9)(1) = 58.5

Is this the most volume we can get?

9 in

6.5 in

58.5 in3

H W of base L of base Volume

x 11 – 2x 8.5 – 2x y =

Homework Quiz
• Get out your graphing calculator
How do we find the “best” box?

http://www.mste.uiuc.edu/users/carvell/3dbox/

• We use algebra and our calculator to find THE x that will give us the dimensions for the MOST volume.
• Go to Y= on your calculator and enter our volume equation: y = x(8.5 – 2x)(11 – 2x)
• For our data, we must set our Window to see the part of the graph we need.
xmin:0

square dimension & height of prism

can’t have negatives

xmax:15

xscl: 1

ymin: 0

volume

ymax: 100

yscl: 10

xres: 1

Then graph

Trace to find max volume (y) on graph

Use Calc (2nd trace) to find exact max.

left bound is (0,0)

right bound?

What x do we get?

Window and Steps
Finally . . .
• The maximum volume for a rectangular prism made from an 8.5 x 11 sheet of paper will be 66.148 in3 and will occur when the squares cut from each corner measure 1.5854 in x 1.5854 in.
• How much is that? Let’s eat!
Classwork and Homework
• Page 532 # 1, 2, 3, 5, 10 (collecting tomorrow)
• Study for your test tomorrow.
• naming prisms, pyramids, cylinders, cones and their parts (10.1)
• finding volumes of those objects (10.2, 10.3)
• doing an application problem (10.4)