Calculus Optimal Box Cost Question

Question

I'm self teaching myself from a popular textbook for fun, the solutions are vague.

Here's the problems.

Cost of materials:

1.) Lid = ($2r$) per square unit

2.) Sides & bottom = ($r$) per square unit

Find the dimensions that of the box that minimze the total cost.

(Deleted)

Additional info:

V=hxy y=V/xh (Cost)=C=3rxy + 2rxh + 2ryh ΔC/ΔX=2rh-2rV/x^2=0 x^2=V/h x=y

I just realized that I added the solution from a different problem. My mistake.

Answer 1

For dimension $l,b,h$,

$$V=lbh$$

$$C=(2lb+lb+2(bh+lh))*r$$

By Lagranges' method,

$$c=(2lb+lb+2(bh+lh))*r+k(V-lbh)$$ Take partial derivatives of $c$ w.r.t. $l,b,h$ and express in terms of $k$ and solve for $k$ by substituting in $V-lbh=0$ equation.