Question

The volume (in cubic meters), vv, of a rectangular room is given by the expression: v = 162-2b^6v=162?2b 6 Where bb is a positive integer and each dimension is an integer greater than 11 meter. What are three unique expressions that could represent the dimensions of the room in terms of bb ?

Answer 1

ANSWER

The 3 unique dimensions are


$2 \times ( 9- {b}^{3}) \times ( 9 + {b}^{3} )$


EXPLANATION

The expression for the volume of the rectangular room is

$V = 162 - 2 {b}^{6}$
where is an integer greater than 1.


We factor the HCF to obtain,




$V = 2 (81- {b}^{6} )$


We rewrite the expression in the parenthesis as difference of two squares to obtain,


$V = 2 ( {9}^{2} - ( {b}^{3} )^{2} )$



Recall that,


${a}^{2} - {b}^{2} = (a + b)(a - b)$


This implies that,



$V = 2 ( 9- {b}^{3})( 9 + {b}^{3} )$