Answer
Dimensions of a box with volume 1000 and minimal surface area must be $a=b=c=10 cm$
Work Step by Step
Volume of a box is given by $V=abc=1000 cm^3$
Surface area, $S=2ab+2bc+2ca$
This implies $\nabla V=\lt ab,bc,ca \gt$ and
$\nabla S=\lt 2b+2c,2c+2a,2a+2b \gt $
After solving we get, $a=b=c=10 cm$
Hence, the dimensions of a box with volume 1000 and minimal surface area must be $a=b=c=10 cm$
