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THE POPCORN PROBLEM
Equal squares are cut from the four corners of a sheet of paper which measures 21 cm x 28 cm and a box is made by folding up the flaps. What is the side of the square that must be cut out to create a box with the greatest volume?
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1. How many different boxes are possible is the squares must have sides measured in whole centimeters?
2. Your group will be constructing two different boxes. Write down the length of the side of the squares here.
Box 1:________ Box 2:________
3. Compare boxes with your classmates to find the box with the greatest volume. What is the length of the side of the square of the box with the greatest volume?
4. Measure the height, length, and width of each of your boxes and find the volume.
Box 1 Box 2
Height:________ Height:________
Length:________ Length:________
Width:________ Width:________
Volume:________ Volume:________
5. Record the volumes for every possible box. In the last row, find an expression for the length, width, and volume of the box.
HeightLengthWidthVolumex6. Graph the height of the box on the xaxis and the volume of the box on the yaxis on a separate sheet of graph paper.
7. Based on the graph and your table of values, what do you think is the exact value of x that will maximize the volume of the box? Explain.
8. Enter the height and volume of the box into two lists on a graphing calculator. Which regression equation should be used? Find the regression equation of the data.
9. Find the derivative of this equation.
10. Set this equal to zero and solve for x using the quadratic formula. How does this answer compare to the answer to #7?
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