(1) From the statement, we know that:
• the volume of a rectangular prism is:
[tex]V(x)=48x^3+56x^2+16x\text{.}[/tex]
• and it also can be computed by:
[tex]V=a\cdot b\cdot c.[/tex]
Where a, b and c are the lengths of the sides.
By factoring in the expression of V(x), we get:
[tex]\begin{gathered} V(x)=(48x^3+56x^2+16x) \\ =(8x)\cdot(6x^2+7x+2) \\ =(8x)\cdot(3x+2)\cdot(2x+1)\text{.} \end{gathered}[/tex]
Comparing this result with the expression above, we see that the sides of the prism are:
[tex]\begin{gathered} a=8x, \\ b=3x+2, \\ c=2x+1. \end{gathered}[/tex]
(2) Relacing the value x = 2 in the polynomial V(x), we get:
[tex]V(2)=48\cdot2^3+56\cdot2^2+16\cdot2=640.[/tex]
(3) Replacing the value x = 2 in the expressions for the dimensions found in point (2), we get:
[tex]\begin{gathered} a=8\cdot2=16, \\ b=3\cdot2+2=8, \\ c=2\cdot2+1=5. \end{gathered}[/tex]
(4) Replacing the dimensions from point (3) in the expression for the volume, we get:
[tex]V=16\cdot8\cdot5=640.[/tex]
Which is the same result that we get in point (2), as it should be.
Answer
(1) Dimensions
[tex]\begin{gathered} a=8x, \\ b=3x+2, \\ c=2x+1. \end{gathered}[/tex]
(2) Volume when x = 2:
[tex]V(2)=640.[/tex]
(3) Dimensions when x = 2:
[tex]\begin{gathered} a=16, \\ b=8, \\ c=5. \end{gathered}[/tex]
(4) Volume when x = 2:
[tex]V=16\cdot8\cdot5=640.[/tex]