Answer:
[tex]32000\ \text{cm}^3[/tex]
Step-by-step explanation:
b = Length and breadth of base
h = Height of box
Surface area of box = [tex]4800\ \text{cm}^2[/tex]
Volume is given by
[tex]V=b^2h[/tex]
Surface area is given by area of base plus the area of the four sides
[tex]b^2+4hb=4800\\\Rightarrow h=\dfrac{4800-b^2}{4b}\\\Rightarrow h=\dfrac{1200}{b}-\dfrac{b}{4}[/tex]
[tex]V=b^2h=b^2(\dfrac{1200}{b}-\dfrac{b}{4})\\\Rightarrow V=1200b-\dfrac{b^3}{4}[/tex]
Differentiating with respect to the base dimensions we get
[tex]\dfrac{dV}{db}=1200-\dfrac{3}{4}b^2[/tex]
Equating with 0
[tex]0=1200-\dfrac{3}{4}b^2\\\Rightarrow b=\sqrt{\dfrac{1200\times 4}{3}}\\\Rightarrow b=40\ \text{cm}[/tex]
Finding the double derative
[tex]\dfrac{d^2V}{db^2}=-\dfrac{3}{2}b[/tex]
at [tex]b=40[/tex]
[tex]\dfrac{d^2V}{db^2}=-\dfrac{3}{2}b=-\dfrac{3}{2}\times 40=-60\\ -60<0[/tex]
So, the maximum value of b is 40 cm
[tex]h=\dfrac{1200}{b}-\dfrac{b}{4}=\dfrac{1200}{40}-\dfrac{40}{4}\\\Rightarrow h=20\ \text{cm}[/tex]
Volume is given by
[tex]V=b^2h=40^2\times20\\\Rightarrow V=32000\ \text{cm}^3[/tex]
The largest possible volume of the box is [tex]32000\ \text{cm}^3[/tex]
