A rectangular box must have a volume of $2$ cubic meters. The material for the base and top costs $$2$ per square meter. The material for the vertical sides costs $$8$ per square meter.

$($a$)$ Express the total cost of the box in terms of the length $($l$)$ and width $($w$)$ of the base.

Cost$=$$$43$Lw$2$

$($b$)$ Find the dimensions of the box that costs least.

length$=2$ meters

width$=2$ meters

height$=12$ meters