Hello, leinadwerdna!

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:

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/ | / | H
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H | | /
| |/ W
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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?