Applications of Cubic Functions by pptfiles

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									Applications of Cubic
      Functions
   Volume of a Open Box.
Suppose you are trying to make an open-top box out
of a piece of cardboard that is 20 inches by 16
inches. You are to cut the same size square from
each corner. Write a function to represent the
volume of this box.
                      20
               x                  x
       x                                        x
                   20 - 2x




                                      16 - 2x

                                                    16
       x                                        x

           x                      x
              V=lwh

     V  (20  2 x)(16  2 x)(x)
20 - 2x




                               x
                               ?

                              16 - 2x
                    20 - 2x
Formula for the Volume of a Box

V  ( L  2 x)(W  2 x)(x)
   The final answer for the volume will
        ALWAYS have the term :


                        3
             4x
Write the formula for the volume of
              our box:

    V  (20  2 x)(16  2 x)(x)
    Step 1: Multiply the two binomials together
    Step 2: Multiply by x
       16        -2x
                            V  (4 x  72 x  320 ) x
                                    2

   20 320       -40x


         -32x   4x2
                            V  4 x  72x  320x
                                    3       2

   -2x
What is the maximum volume?
• What is the possible domain for this box?
   What is the greatest possible value that
             we can cut out for x?
• 0 < X < 8 (Half of the length of the smallest side)
  • SO, Xmin = 0 and Xmax = 8; ZOOM 0
           • Do you want x or y?
                      • Y!!!
             • 420 cubic inches
What size square should be cut
from each corner to realize the
      maximum volume?

   • What do you want now?
             • X!!
         • 2.9 inches
What size square should you cut
 from each corner to realize a
 volume of 300 cubic inches?
      • What do you know: x or y?
 • Y!! Let y = 300; find the intersection
        • 1.3 inches or 5 inches
What is the volume if a square with
 side 2 inches is cut from each
              corner?
      • What do you know; x or y?
                  • X!!!
         • Go to table; let x = 2
           • 384 Cubic inches

								
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