Question

Use LaGrange multipliers to solve the following problem: A closed box is to be constructed with volume 750in^3. The material for the top and bottom costs 3 cents per square inch, the material for the front and back costs 6 cents per square inch, and the material for two ends costs 9 cents per square inch. What dimensions should the box have to minimize the cost of the construction materials? What is the minimum cost per box?

Answer 1

Here are your equations and constraints.
\[V=xyz=750\]
\[A=2xy+2xz+2yz\]
\[C=3(2xy)+6(2xz)+9(2yz)\]

Answer 2

The cost \(C\) is minimized when the amount of material \(A\) is minimized. Lagrange multipliers gives
\[\begin{cases}\nabla A(x,y,z)=\lambda\nabla V(x,y,z)\\
V(x,y,z)=750\end{cases}\]
or
\[\begin{cases}
2y+2z=\lambda yz\\
2x+2z=\lambda xz\\
2x+2y=\lambda xy\\
xyz=750
\end{cases}\]
You need to find \(x,y,z\) that meet these conditions.

Answer 3

Ok so now just solve the system of equations and plug the answers into the original equation to find the minimum?

Answer 4

Yes. You can assume \(\lambda\not=0\).