A closed rectangular box with a volume of $16000$ $\text{cm}^3$ is made from two kinds of materials. The top and bottom are made of material costing $\$0.10$ per centimetre squared and the sides from material costing $\$0.05$ per centimetre squared. Find the dimensions of the box so that the cost of materials is minimized.
I have $V(x,y,z)=16000$ and Cost $C(x)=0.2xy+0.1zx+0.1zy$
Using $F(x,y,z,\lambda)=0.2xy+0.1zx+0.1zy+\lambda(xyz-16000)$ We get the system:
$F_x= 0.2y+0.1z+\lambda(yz) = 0$
$F_y= 0.2x+0.1z+\lambda(xz) = 0$
$F_z=0.1y+0.1x+\lambda(xy) = 0$
$F_\lambda=xyz-16000 = 0$
Taking first and second and then second and third equations,
I get $x=y$
$y=20$
$2y=z = 40$
$\lambda=-4.005$
I just want to make sure I did the process correctly, and that the process of maximizing and minimizing are the same, but just depend on the value of $\lambda$.
thank you and any pointers are appreciated.