Answer
Length = 2 in
Width = 15 in
Height = 6 in
Work Step by Step
Let $x$ ft be the height of the box.
Based on the given conditions, we have:
$Volume=length\times width\times height$
$180=(x-4)x(x+9)$
Solve for $x$:
$(x-4)x(x+9)=180$
$x^3+5x^2-36x=180$
$x^3+5x^2-36x-180=0$ (Factorize)
$(x+6)(x+5)(x-6)=0$
$x=-6$ or $x=-5$ or $x=6$
Since $x$ must be positive, the solution is $x=6$.
Therefore,
(i) The length is $x-4=6-4=2$ in
(ii) The width is $x+9=6+9=15$ in
(iii) The height is $x=6$ in
