Let w represent width of the rectangle.
We have been given that the length of a rectangular prism is 1 more than three times the width w. So length of the rectangle would be [tex]3w+1[/tex].
We have been given that the volume of the prism is [tex]3w^3+19w^2+6w[/tex].
We know that volume of rectangular prism is width times length times height.
So we can set an equation as:
[tex]\text{Length}\times \text{Width}\times\text{Height}=3w^3+19w^2+6w[/tex]
[tex]w(3w+1)\times\text{Height}=3w^3+19w^2+6w[/tex]
[tex]\text{Height}=\frac{3w^3+19w^2+6w}{w(3w+1)}[/tex]
Let us factor out w from numerator.
[tex]\text{Height}=\frac{w(3w^2+19w+6)}{w(3w+1)}[/tex]
[tex]\text{Height}=\frac{3w^2+19w+6}{3w+1}[/tex]
[tex]\text{Height}=\frac{3w^2+18w+w+6}{3w+1}[/tex]
[tex]\text{Height}=\frac{3w(w+6)+1(w+6)}{3w+1}[/tex]
[tex]\text{Height}=\frac{(w+6)(3w+1)}{3w+1}[/tex]
Now we will cancel out [tex](3w+1)[/tex] from numerator and denominator.
[tex]\text{Height}=w+6[/tex]
Therefore, the height of the prism is [tex]w+6[/tex] units.
