Answer:
The possible length and width of the prism are [tex](x-4)m[/tex] and [tex](x^2+4x+16)m[/tex].
Step-by-step explanation:
The base of the rectangular prism is rectangular. Let [tex]l[/tex] and [tex]w[/tex] be the length and width of the base.
Area of the base[tex]= lw[/tex]
Given that the volume,
[tex]V=(x^4+ 5x^3– 64x - 320) m^2[/tex],
and the height,
[tex]h= (x+5) m[/tex].
The volume of the prism = (Area of the base) x (Height).i.e.
[tex]V=lwh[/tex]
[tex]\Rightarrow lw =\frac{V}{h}[/tex]
[tex]\Rightarrow lw =\frac{(x^4+ 5x^3– 64x - 320)}{(x+5)}[/tex]
[tex]\Rightarrow lw =\frac{(x+5)(x^3-64)}{(x+5)}[/tex]
[tex]\Rightarrow lw =x^3-64[/tex]
[tex]\Rightarrow lw =x^3-4^3[/tex]
[tex]\Rightarrow lw =(x-4)(x^2+4x+16)[/tex] [ using the identity [tex]p^3-r^3=(p-r)(p^2+pr+r^2)][/tex]
Hence, the possible length and width of the prism are[tex](x-4)m[/tex] and [tex](x^2+4x+16)m[/tex].
