1. Minizing Cost

Lagrangeco, inc. has to build a rectangular steel box with an open top of volume 96 cm3.
The bottom of the box must be reinforced and costs three times as much per square cm as
the four sides. What dimensions of the box will minimize the cost?

I realize the constraint is lwh=96 but how do you write the cost function

Did you make a sketch?

Lagrangeco, inc. has to build a rectangular steel box with an open top of volume 96 cm³.
The bottom of the box costs three times as much per cm² as the four sides.
What dimensions of the box will minimize the cost?
Code:
         * - - - - *
/|        /|
/ |       / | H
* - - - - *  |
|         |  *
H |         | /
|         |/ W
* - - - - *
L

The cost of the steel is $\displaystyle k$ dollar/cm² for the sides
. . and $\displaystyle 3k$ dollars/cm² for the bottom.

The bottom has $\displaystyle LW$ cm² of steel at $\displaystyle 3k$ dollars/cm².
. . Its cost is: .$\displaystyle kLW$ dollars.

The four sides has $\displaystyle 2LH + 2WH$ cm² of steel at $\displaystyle k$ dollars/cm².
. . Their cost is: .$\displaystyle 2(L+W)H\cdot k \:=\:2k(L+W)H$ dollars.

The total cost is: .$\displaystyle C \;=\;3kLW + 2k(L+W)H$ dollars.

Got it?

3. i took the gradient of the cost function and the constraint and then set each component = to the respective component times lamda but im not sure how to solve the system of equations could u help me

4. i need some help with this