Given a rectangular box with an open top and square base, the dimensions of the box are:
[tex]a\times a\times b[/tex]
The volume can be calculated as:
[tex]V=a\cdot a\cdot b=a^2\cdot b[/tex]
The area of the sides is:
[tex]A_L=a\cdot b[/tex]
The area of the base:
[tex]A_B=a^2[/tex]
There are 4 lateral sides and 1 base (the top is open), so the total surface area is:
[tex]A_{\text{total}}=4\cdot A_L+A_B=4\cdot a\cdot b+a^2[/tex]
We have a fixed volume of 2048 in³, then:
[tex]\begin{gathered} a^2\cdot b=2048 \\ b=\frac{2048}{a^2} \end{gathered}[/tex]
Using this result on A_total:
[tex]A_{\text{total}}=4\cdot a\cdot\frac{2048}{a^2}+a^2=\frac{8192}{a}+a^2[/tex]
To find the minimum surface area, we take the derivative:
[tex]\begin{gathered} \frac{dA_{total}}{da}=-\frac{8192}{a^2}+2a=0 \\ a^3=4096 \\ a=16 \end{gathered}[/tex]
Now, we calculate the minimum total area using a:
[tex]A_{\text{total}}=\frac{8192}{16}+16^2=768in^2[/tex]
