1. Maximizing Volume March 23, 2004

2. Profit and Product Distribution... Why does cereal come in a rectangular prism and potato chips come in a bag?

3. 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.

4. 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?

5. 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? x = 1

6. 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?

7. Group Activity: Use your paper to cut and then fill in this table. H W of base L of base Volume 1 in 1.5 in 2 in 3 in 4 in 9 in6.5 in58.5 in 3 x11 – 2x8.5 – 2x y =

8. Homework Quiz 1. Get out your chart 2. Get out your box 3. Get out your graphing calculator

9. How do we find the “best” box? http://www.mste.uiuc.edu/users/carvell/3dbox/ Add a surface area column to your chart! 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.

10. Window and Steps 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 (2 nd trace) to find exact max. – left bound is (0,0) – right bound? What x do we get?

11. Finally... The maximum volume for a rectangular prism made from an 8.5 x 11 sheet of paper will be 66.148 in 3 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!

12. 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) – about 15 problems