Answer
The dimensions of a box with volume 1000 and minimal surface area must be $a=b=c=10 cm$.
Work Step by Step
Need to apply Lagrange Multipliers Method to determine the maximum volume of a rectangular box.
we have $\nabla f=\lambda \nabla g$
The volume of a rectangular box is $V=abc$
As per question, we have $V=abc=1000 cm^3$
and Surface area, $S=2ab+2bc+2ca$
Now, $\nabla V=\lt ab,bc,ca \gt$ and
$\nabla S=\lt 2b+2c,2c+2a,2a+2b \gt $
Simplify to get the values of a,b and c.
We have $a=b=c=10 cm$
The dimensions of a box with volume 1000 and minimal surface area must be $a=b=c=10 cm$.
