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Question

Question #1

 

A bottom for a box can be made by cutting congruent squares from each of the four corners of a piece of cardboard. The volume of the box made from 8.5- by - 11- inch piece of cardboard would be represented by V(x)=x(11-2x)(8.5-2x), where x is the length of one square. Find the volume when x = 1 inch 

 

Question #2

Find an expression for the width of a rectangle whose length is represented by x-2 and whose are is represented by

2x³-8x²+ 2x+12

Answer 1

1)  You already have the formula:

     V(x) = x(11-2x)(8.5-2x)

     To find the volume when x=1 inch

     substitute 1 in place of each x in the equation

 

v(1) = 1(11-2)(8.5-2)

v(1) = 9(6.5)

v(1) = 58.5 in³ 

 

2)  L = x-2

     Area of a rectangle is length times width

     A = LW

     Solving this for width:

     W = A/L

 

width = (2x³-8x²+2x+12)/(x-2)

We can do this by long division

         2x² -4x -6

      ___________________

x-2 |  2x³ - 8x² + 2x + 12    

       -(2x³-4x²)

       -------------------------

                -4x² + 2x + 12

             -(-4x² + 8x)

             -------------------

                       -6x + 12

                    -(-6x + 12)

                     -------------

                                 0

 

Width = 2x² - 4x -6

 

*************************************

                

You could also us synthetic division if familiar.

If (x-2) is  a direct factor of  2x³ - 8x² + 2x + 12    

we will have no remainder when using

2 for the synthetic factor

 

 

2 | 2    -8    2     12

            4   -8    -12

    --------------------

     2    -4    -6     0

 

Since the remainder is 0, this is a direct factor and

your answer is  W = 2x²-4x-6

 

This is the same answer obtained via long division