1. example 2 Finding the Maximum Volume Chapter 6.3 A box is to be formed by cutting squares x inches per side from each corner of a square piece of cardboard that is 24 inches on each side and folding up the sides. This will give a box whose height is x inches, with the length of each side of the bottom 2x less than the original length of the cardboard. Using V = lwh, its volume is given by where x is length of the side of the square that is cut out.  2009 PBLPathways

2. A box is to be formed by cutting squares x inches per side from each corner of a square piece of cardboard that is 24 inches on each side and folding up the sides. This will give a box whose height is x inches, with the length of each side of the bottom 2x less than the original length of the cardboard. Using V = lwh, its volume is given by where x is length of the side of the square that is cut out.

3.  2009 PBLPathways a.Use the factors of V to find the values of x that give volume 0. b.Graph the function that gives the volume as a function of the side of the square that is cut out over an x-interval that includes the x-values found in part (a). c.What input values make sense for this problem (that is, actually result in a box)? d.Graph the function using the input values that make sense for the problem (found in part (c)). e.Use technology to determine the size of the squares that should be cut out to give the maximum volume.

4.  2009 PBLPathways a.Use the factors of V to find the values of x that give volume 0.

5.  2009 PBLPathways a.Use the factors of V to find the values of x that give volume 0.

6.  2009 PBLPathways a.Use the factors of V to find the values of x that give volume 0.

7.  2009 PBLPathways a.Use the factors of V to find the values of x that give volume 0.

8.  2009 PBLPathways a.Use the factors of V to find the values of x that give volume 0.

9.  2009 PBLPathways b.Graph the function that gives the volume as a function of the side of the square that is cut out over an x-interval that includes the x-values found in part (a). x V

10.  2009 PBLPathways b.Graph the function that gives the volume as a function of the side of the square that is cut out over an x-interval that includes the x-values found in part (a). x V

11.  2009 PBLPathways b.Graph the function that gives the volume as a function of the side of the square that is cut out over an x-interval that includes the x-values found in part (a). x V

12.  2009 PBLPathways c.What input values make sense for this problem (that is, actually result in a box)? All dimensions must be positive.

13.  2009 PBLPathways c.What input values make sense for this problem (that is, actually result in a box)? All dimensions must be positive.

14.  2009 PBLPathways c.What input values make sense for this problem (that is, actually result in a box)? All dimensions must be positive.

15.  2009 PBLPathways c.What input values make sense for this problem (that is, actually result in a box)? All dimensions must be positive.

16.  2009 PBLPathways c.What input values make sense for this problem (that is, actually result in a box)? All dimensions must be positive.

17.  2009 PBLPathways c.What input values make sense for this problem (that is, actually result in a box)? All dimensions must be positive.

18.  2009 PBLPathways d.Graph the function using the input values that make sense for the problem (found in part (c)). x V

19.  2009 PBLPathways d.Graph the function using the input values that make sense for the problem (found in part (c)). x V

20.  2009 PBLPathways e.Use technology to determine the size of the squares that should be cut out to give the maximum volume. x V

21.  2009 PBLPathways e.Use technology to determine the size of the squares that should be cut out to give the maximum volume. x V (4, 1024)

22.  2009 PBLPathways e.Use technology to determine the size of the squares that should be cut out to give the maximum volume. x V (4, 1024) 4 16