SOLUTION: A closed rectangular container with a square base is to have a volume of 18000 inches^3 . The material of the top and bottom of the container will cost $2 per inch^2 , the material

Let x be the length of the side of the base, and

let y be the height of the container.


Then the volume is   = 18000 cubic inches.


The cost of the material for two bases is   dollars;

the cost of the material for four lateral sides is  4*3*x*y = 12xy dollars.


The problem requires us to minimize   = 18000.


Using the restriction formula, express  y =   and substitute it  into the total cost expression.


Then the cost takes the form  C(x) =  = .


Thus the problem is just reduced to finding minimum of the function


    C(x) = .


To find it minimum, take a derivative and equate it to zero.  It gives you the equation

    C'(x) =  = 0,

which implies

    4x^3 = 216000,

     x^3 = 54000

     x =  = .


Then  y =  =  = .


ANSWER.  x=  = 37.798 inches (approximately)  and  y=  = 12.6 inches (approximately).


PARTIAL CHECK.   =  = 18001 in^3.  The miserable difference is due to rounding.