## Calculus 8th Edition

The dimensions of a box with volume 1000 and minimal surface area must be $a=b=c=10 cm$.
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 $a=b=c=10 cm$.