Story problem

Question

A shoe company needs a shoebox that has a volume of 288 cubic inches. If the width of the shoebox needs to be 6 inches less than the length and the height needs to be 8 inches less than the length, determine the dimensions of the shoebox

Answer 1

Lwh= 288

w = L-6

h = L-8

 

L(L-6)(L-8) = 288

L(L²-14L + 48) = 288

L³ - 14L² + 48L - 288 = 0 

 

The possible real roots of the polynomial will be factors of -288

±{1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288}

 

With synthetic division trying possible roots I found:

 

12 |  1    -14    48    -288

    |          12   -24     288

     -----------------------------

       1      -2     24     |0

                                -----

 

Since the remainder is zero, L=12 is a root and (L-12) is a factor

 

(L-12)(L² - 2L + 24) = 0

 

Using L = 12 we have:

 

length 12 inches

width 12-6 = 6 inches

height 12-8 = 4 inches

volume 12(6)(4) = 288 in³

 

 

From the L²-2L+24 and using quadratic equation

we find the other roots are complex and thus discardable