Minimize Total Cost of Box So there is a rectangular box that has a volume of $8 m^3$. The top and bottom of the box is made with some material that has a cost of $8$ dollars per square meter. The sides are made with another material that costs $1$ dollar per square meter. How can I find the dimensions of the box that would minimize the total cost?
My thoughts:
I first started off with the equation, $V=lwh$. I think that we need to find an equation for cost in which we substitute $h$ into and then solve for two variables, then use the first derivative. I can get the concept but can't get how to work on the mechanics.


Let $a$ and $b$ denote the horizontal sides of the rectangular box and $c$ its height. One seeks to solve the following problem:
\begin{align*}
\min_{(a,b,c)\in\mathbb R_+^3}&\left\{8(2ab)+1(2ac+2bc)\right\}\\\text{s.t.}\quad&\,abc=8.
\end{align*}
Substitute for $c$ using the constraint $c=8/(ab)$ and plug it into the minimand to get, after some rearrangement: $$16 ab+\frac{16}{b}+\frac{16}{a}.$$ I leave it to you to check that this function is minimized at $a=b=1$ (note that it is now an unconstrained minimization problem). Hence, the optimal solution is $$(a^{\star},b^{\star},c^{\star})=(1,1,8).$$
