Question

The length of a rectangular prism is 1 more than three times the width w. The volume of the prism is 3 w cubed plus 19 w squared plus 6 w. What is a simplified expression for the height of the​ prism?

Answer 1

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 $3w+1$.

We have been given that the volume of the prism is $3w^3+19w^2+6w$.

We know that volume of rectangular prism is width times length times height.

So we can set an equation as:

$\text{Length}\times \text{Width}\times\text{Height}=3w^3+19w^2+6w$

$w(3w+1)\times\text{Height}=3w^3+19w^2+6w$

$\text{Height}=\frac{3w^3+19w^2+6w}{w(3w+1)}$

Let us factor out w from numerator.

$\text{Height}=\frac{w(3w^2+19w+6)}{w(3w+1)}$

$\text{Height}=\frac{3w^2+19w+6}{3w+1}$

$\text{Height}=\frac{3w^2+18w+w+6}{3w+1}$

$\text{Height}=\frac{3w(w+6)+1(w+6)}{3w+1}$

$\text{Height}=\frac{(w+6)(3w+1)}{3w+1}$

Now we will cancel out $(3w+1)$ from numerator and denominator.

$\text{Height}=w+6$

Therefore, the height of the prism is $w+6$ units.