Respuesta :
Dimension of box is 2 m x 2 m x 1.33 m
Step-by-step explanation:
Let a be base side and h be the height.
Volume of box, V = a²h
The sides of the box will cost $3 per m² and the base will cost $4 per m². Cost for making is $48.
That is
         4a² + 3 x 4 x a x h = 48
         4a² + 12 a x h = 48
         a² + 3 ah = 12 Â
         [tex]h=\frac{12-a^2}{3a}[/tex]
So volume is
          [tex]V=a^2\times \frac{12-a^2}{3a}=4a-\frac{a^3}{3}[/tex]
At maximum volume we have derivative is zero,
          [tex]dV=4da-3\times \frac{a^2}{3}da\\\\0=4-a^2\\\\a=\pm 2[/tex]
Negative side is not possible, hence side of square base is 2m.
Substituting in  a² + 3 ah = 12 Â
              2² + 3 x  2 x h = 12 Â
               h = 1.33 m
Dimension of box is 2 m x 2 m x 1.33 m
