Question 538760
We  want to know the length of the sides of that square bottom.
Let that length (in cm) be s.
The surface of the bottom (in square cm) is {{{s^2}}}.
The surface of the top is the same.
The surface of the side walls (in square cm) equals the height in cm (5) times the perimeter of the bottom in cm (4s).
The total surface area is
{{{2(s^2)+5(4s)= 2s^2+20s}}}
That gives us the equation
{{{2s^2+20s=192}}} ---> {{{s^2+10s=96}}} ---> {{{s^2+10s-96=0}}}
We solve that equation any way we can.
Factoring, {{{s^2+10s-96=(s+16)(s-6)}}}, so the equation can be written as
{{{(s+16)(s-6)=0}}}, which has the solutions {{{s=6}}} and {{{s=-16}}}.
Since the measurements of the box must be positive numbers, the only reasonable answer is {{{s=6}}}.
The volume is {{{(6cm)(6cm)(5cm)=180cm^3}}}