Answer:
[tex]\mathsf{height}=x^2+x-9[/tex]
Step-by-step explanation:
Volume of a cuboid = L x w x h
(where L is the length, w is the width and h is the height)
[tex]\implies \mathsf{height=\frac{volume}{width \times length} }[/tex]
Given:
[tex]\mathsf{volume}=3x^4-3x^3-33x^2+54x[/tex]
[tex]\mathsf{width}=x-2[/tex]
[tex]\mathsf{length}=3x[/tex]
[tex]\implies \mathsf{height}=\frac{3x^4-3x^3-33x^2+54x}{3x(x-2)}[/tex]
Factor expression for volume
Factor out common term [tex]3x[/tex]: [tex]3x(x^3-x^2-11x+18)[/tex]
Factor [tex]x^3-x^2-11x+18[/tex]: [tex]3x(x-2)(x^2+x-9)[/tex]
[tex]\implies \mathsf{height}=\frac{3x(x-2)(x^2+x-9)}{3x(x-2)}[/tex]
Cancel the common factors [tex]3x(x-2)[/tex]:
[tex]\implies \mathsf{height}=x^2+x-9[/tex]
