Question

The volume (in cubic meters), v, of a rectangular prism is given by the expression: v = a^8 - b^8, equals, a, start superscript, 8, end superscript, minus, b, start superscript, 8, end superscript where aaa and bbb are positive integers. pick three expressions that together can represent the three dimensions of the prism (each in meters).

Answer 1

The three dimensions of the prism are:

$Length= (a^4+b^4)meter, Width= (a^2 +b^2)meter, Height=(a^2-b^2)meter$

Explanation

The volume (in cubic meters), $v$ , of a rectangular prism is given by the expression:  $v = a^8 - b^8 .......................(1)$  

Formula for the volume of a rectangular prism:  $v= x*y*z ..................(2)$

where $x, y, z$ are the length, width and height of the rectangular prism.

Now comparing equation (1) and (2) , we will get....

$x*y*z= a^8 -b^8$

By factoring out the right side...

$x*y*z= (a^4)^2 -(b^4)^2 \\ \\ x*y*z= (a^4 + b^4)(a^4 - b^4)\\ \\ x*y*z= (a^4 + b^4)[(a^2)^2 - (b^2)^2]\\ \\ x*y*z= (a^4+ b^4)(a^2+ b^2)(a^2 - b^2)$

After comparing left and right side, we can say...

$x= a^4 + b^4 , y= a^2 + b^2 , z= a^2 -b^2$

So, the three dimensions of the prism are:

$Length= (a^4+b^4)meter, Width= (a^2 +b^2)meter, Height=(a^2-b^2)meter$