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 * ( 9- {b}^(3)) * ( 9 + {b}^(3) )`


Step-by-step 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) )`