Application of Quadratics

Question

A Sheet of cardboard 42cm long and 24cm wide has squares cut out of the corners so it can be folded to form an open box. Find the dimensions of the box if the volume of the box is 468cm^3. Using Quadratics.

Answer 1

Let x = length of a side of the square

 

Then, length of box = 42-2x

  

         width of box = 24-2x

 

         height of box = x

 

Since volume = 468, we have x(42-2x)(24-2x) = 468

 

                                            x(4x²-132x+1008) = 468

 

                                            4x³-132x²+1008x-468 = 0

 

                                             x³-33x²+252x-117 = 0

 

By the Rational Zero Theorem, the only possible rational roots are 1, 3, 13, 39, and 117.  Since 0<x<12 (if x≥12, then the width would be negative), only 1 and 3 need to be considered.  However, neither 1 nor 3 works, so x must be irrational.  You can approximate the root on a graphing calculator or by using Wolfram Alpha, for example.