SOLUTION: Equipment - MathResources Graphing Tool (provided) - 8 pieces of 8.5 x 11 paper - Scissors - Ruler - Tape - Pencil Procedure 1. Cut each of the 8 pieces of paper into squ

Question 206899: Equipment
- MathResources Graphing Tool (provided)
- 8 pieces of 8.5 x 11 paper
- Scissors
- Ruler
- Tape
- Pencil
Procedure
1. Cut each of the 8 pieces of paper into squares measuring 8 inches by 8 inches.
2. Using a pencil and ruler, mark each side off in 1/2 inch increments so that you have created a piece of graph paper that has 16 squares by 16 squares. Repeat this procedure for all 8 pieces of paper.
3. Take one sheet of graph paper and cut out one square out of each of the 4 corners. Fold up the 4 sides to create a box (without a lid) that has 14 squares across the bottom and has a height of 1 square up. You may want to tape the sides in place. This is your first of 8 boxes.
4. Repeat step #3 cutting 1 inch squares from each of the 4 corners and folding up the sides. Continue with the remaining 6 pieces of paper increasing the size of the squares that you will cut out by 1/2 inch. Can the last piece of paper have 4 inch squares cut from each corner? Why or why not?
5. Create a data table like the example data table below.
Note: The measurements are in inches, not squares. Each square is 1/2 inch. Volume should be given in cubic inches.

6. Find the length, width, height, and volume (V = lwh) of each of the 7 boxes, to complete the data table.
Questions
1. Which box has the maximum volume?
2. With the MathResources Graphing Tool, graph the data by placing the height on the horizontal axis and the volume on the vertical axis.) Use a range of x from -5 to 10 and y from -5 to 40. Title and label the axes of your graph. To graph the data click on the button below.

To see your graph, click on the pencil icon. I would recommend pasting the graph into a Word document. Then you can view it throughout the rest of the activity. Choose File, Image, Copy from the menu and then just paste it into a document that you can continuously view.
Very Important: When the Properties dialog pops up choose the "cubic" option for the line that best fits your graph. If the line fits, then your data fits a cubic function.
3. What type of function is your graph? Why?
4. What is the equation of your function? To find this, click on the key icon. You will have to double click inside of the description box and use the right arrow key in order to see the whole equation. Copy the equation from the description box or record the equation on a piece of paper.
5. Use the function plot MathResources Graphing Tool below to graph the function. You can either paste the equation into the function box (remember to get rid of the y = part of the equation) or you can just type in the equation that you recorded in part 4. Use a range of x from -5 to 10 and y from -5 to 40. Again, I would recommend pasting the graph into a Word document. It will be easier to compare the two graphs and easier to save and submit them.

6. Where does your function graph intersect the x-axis? Use the tracker icon to get the coordinates. Round your answers to 2 decimal places and remember, y=0 or close to it. These points are called the zeros of the function. Notice that you are able to see the zeros in the graph of the function but not in the graph of your actual data. What is the reason for that? Think about the differences between real data and the graph of an equation.
7. Locate the highest point on your graph using the tracker icon. What does this point represent?
8. Just for fun, what type of product could be packaged in the container that has the maximum volume? Why?

I need help on this. I know its alot but I have nobody else to help me.
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
If I do all the paper cutting, you won't be able to see it anyway.
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The volume of the box is:
V = x*(16-2x)*(16-2x)
The function intersects the x-axis at x = 0 and x = 8.
The max is at x = 8/3, and the max volume is:
V = (8/3)*(32/3)*(32/3)
Vmax = 8192/27 cubic units =~ 303.4 cubic units
Since the units are 1/2", that's 1024/27 cubic inches = ~37.926 cubic inches
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Notice that the volume is a max when all sides are equal, ie, a regular cube.
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If you have any questions, email me at gsihoutx@aol.com
You can make a nice graph if you dl the FREE software at
http://www.padowan.dk.com/graph/
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