Answer
$${\text{The container is cubic of side }}\root 3 \of {100} $$
Work Step by Step
$$\eqalign{
& {\text{Considering the base of the cardboard drink containers square and}} \cr
& {\text{side }}x,{\text{ its area is given by }}{x^2},{\text{ then te volume of the containers is}} \cr
& V = h{x^2} \cr
& {\text{We know that the container will hold 100c}}{{\text{m}}^3},{\text{ then}} \cr
& 100 = h{x^2}\,\,\,\,\left( {\bf{1}} \right) \cr
& {\text{The area of material used can be represented by}} \cr
& A = 2{x^2} + 4xh\,\,\,\left( {\bf{2}} \right) \cr
& {\text{From the equation }}\left( {\bf{1}} \right){\text{ }}h = \frac{{100}}{{{x^2}}},{\text{ substituting in the equation }}\left( {\bf{2}} \right) \cr
& A = 2{x^2} + 4x\left( {\frac{{100}}{{{x^2}}}} \right) \cr
& A = 2{x^2} + \frac{{400}}{x},{\text{ where }}x > 0 \cr
& {\text{From the graph using GEOGEBRA, we obtain }}x = \root 3 \of {100} \approx 4.641 \cr
& {\text{Then, the dimensions are:}} \cr
& x = \root 3 \of {100} \cr
& h = \frac{{100}}{{{{\left( {\root 3 \of {100} } \right)}^2}}} \cr
& h = \root 3 \of {100} \cr
& {\text{The container is cubic of side }}\root 3 \of {100} \cr} $$
