A rectangular prism's height is x+1. its volume is x^3+7x^2+15x+9. If height and width of the prism are equal, what is its width?

Jan 3, 2017

Width of the prism is $4$ units.

Explanation:

As the volume of a rectangular prism, whose length is $l$, height is $h$ and width is $w$ is $l \times h \times w$.

As the volume of rectangular prism is ${x}^{3} + 7 {x}^{2} + 15 x + 9$,

and height is $\left(x + 1\right)$ and width and height being same, height too is $\left(x + 1\right)$

we can have its length by dividing ${x}^{3} + 7 {x}^{2} + 15 x + 9$ by $\left(x + 1\right) \left({x}_{1}\right) = {x}^{2} + 2 x + 1$.

Dividing ${x}^{3} + 7 {x}^{2} + 15 x + 9$ by $\left({x}^{2} + 2 x + 1\right)$,

$x \left({x}^{2} + 2 x + 1\right) + 5 \left({x}^{2} + 2 x + 1\right) + 4 x + 4$

But as volume is $l \times h \times w$, $4 x + 4 = 4 \left(x + 1\right)$ too should be a multiple of ${x}^{2} + 2 x + 1 = {\left(x + 1\right)}^{2}$,

which is possible if $x + 1 = 4$ i.e. $x = 3$

Hence width is $4$ and height too is $4$

Note that volume is ${3}^{3} + 7 \times {3}^{2} + 15 \times 3 + 9 = 27 + 63 + 45 + 9 = 144$

and length is $\frac{144}{4 \times 4} = 9$.