Question

What is the height, width, and length

Answer 1

so, let's say the length is "L".

width = 2 shorter than L = L - 2.

height = 9 more than twice L = 2L + 9.

check the picture below.

keeping in mind that the volume of a rectangular prism is simply the product of its length, width and height, and since we know the volume has to be 45.

$\bf 45=(L)(L-2)(2L+9)\implies 45=(L)(2L^2+5L-18) \\\\\\ 45=2L^3+5L^2-18L\implies 0=2L^3+5L^2-18L-45 \\\\\\ \stackrel{\textit{now let's do some grouping}}{0=(2L^3+5L^2)-(18L+45)}\implies \stackrel{\textit{common factor}}{0=L^2(2L+5)-9(2L+5)}$

$\bf \stackrel{\textit{again common factor}}{0=\underline{(2L+5)}(L^2-9)}\implies \stackrel{\textit{difference of squares}}{0=(2L+5)\stackrel{\downarrow }{(L^2-3^2)}} \\\\\\ 0=(2L+5)(L-3)(L+3)\implies L= \begin{cases} -\cfrac{5}{2}\\[1em] \boxed{3}\\ -3 \end{cases}$

from those 3 factors, we get those solutions/zeros.

we can't use -5/2, or -3, because the length can't be a negative value, so L = 3.

that means the width = 3 + 2, and the height = 2(3) +9.