Question

Find the dimensions of the box described.

The length is 5 inches more than the width. The width is 2 inches more than the height. The volume is 264 cubic inches.

length ________in

width _________in

height ________in

Answer 1

let the length = x , width = y, height = z

The length is 5 inches more than the width

$x=y+5$

.

The width is 2 inches more than the height

$y=z+2$

and the volume is V=264 cubic inches.

.

volume is given by

Ꮖ ;

V = (y + 5) (2 + 2) 2

264 =(2+2+5)(2+2)

$264=\left(z+7\right)\left(z+2\right)z$

$264=z^3+9z^2+14z$

$z^3+9z^2+14z-264=0$

here we have

$a_0=264,\:\quad a_n=1$

$\mathrm{The\:dividers\:of\:}a_0:\quad 1,\:2,\:3,\:4,\:6,\:8,\:11,\:12,\:22,\:24,\:33,\:44,\:66,\:88,\:132,\:264$

$\mathrm{The\:dividers\:of\:}a_n:\quad 1$

$\mathrm{rational\:numbers:\quad \:}\pm \:\left(\frac{1,\:2,\:3,\:4,\:6,\:8,\:11,\:12,\:22,\:24,\:33,\:44,\:66,\:88,\:132,\:264}{1}\right)$

$\mathrm{rational\:numbers:\quad \:}\pm \:\left(1,\:2,\:3,\:4,\:6,\:8,\:11,\:12,\:22,\:24,\:33,\:44,\:66,\:88,\:132,\:264\right)$

here z=4 is a root so take a synthetic division

3 ܘܕ ܪܘܕ ܢܐ - ܫܒܐ ܀ ܐܐ ܦ ܟ ܪܘܪ- gܐ ܦ ܐ ܬ ܐܐ ܘ S ܘ 13 ܝܠ ܝ ܘ 5 ܘܘ + 13z Z

$z^2+13z+66=0$

find discriminant

$D=b^2-4ac$

$D=13^2-4\cdot \:1\cdot \:66$

$D=169-264$

$D=-95$

here discriminant is negative so expression has complex roots

therefore our correct answer is

${\color{Red} z=4}$

.

$y=z+2$

$y=4+2$

${\color{Red} y=6}$

.

$x=y+5$

$x=6+5$

${\color{Red} x=11}$

.

${\color{Red} length=11\:in,\:\:\:\:width=6\:in,\:\:\:\:\:height=4\:in}$