# acacia's question at Yahoo! Answers regarding using Lagrange Multipliers to minimize cost of a box

#### MarkFL

Staff member
Here is the question:

Find the dimensions of the box which will minimize the TOTAL COST of manufacturing the following open top box of volume 6ft^3?

Solve by the lagrange multiplier method!
Bottom panel costs $3/ft^2 side panel cost$.50/ft^2
Front and back panels cost $1/ft^2 I have posted a link there to this topic so the OP can view my work. #### MarkFL ##### Administrator Staff member Hello acacia, I would orient the box such that the width is$x$, the height is$y$and the length is$z$. Hence, the bottom panel has area$xz$, the side panels have a total area of$2yz$and the front and back panels have a combined area of$2xy$. Let all linear measures be given in feet. Thus, our objective function, the function we wish to minimize is the cost function in dollars, which is given by: $$\displaystyle C(x,y,z)=3xz+yz+2xy$$ Subject to the constraint on the volume: $$\displaystyle g(x,y,z)=xyz-6=0$$ Using Lagrange multipliers, we obtain: $$\displaystyle 3z+2y=\lambda(yz)$$ $$\displaystyle z+2x=\lambda(xz)$$ $$\displaystyle 3x+y=\lambda(xy)$$ Solving for$\lambda$, the first two equations imply: $$\displaystyle \frac{2y+3z}{yz}=\frac{2x+z}{xz}$$ Cross-multiplying, we obtain: $$\displaystyle 2xyz+3xz^2=2xyz+yz^2$$ $$\displaystyle 3xz^2=yz^2$$ Since the constraint requires $$\displaystyle 0<z$$, we may write: $$\displaystyle 3x=y$$ In like manner the first and third equations above imply: $$\displaystyle \frac{2y+3z}{yz}=\frac{3x+y}{xy}$$ Cross-multiplying, we obtain: $$\displaystyle 2xy^2+3xyz=3xyz+y^2z$$ $$\displaystyle 2x=z$$ Substituting for$y$and$z$into the constraint, we obtain: $$\displaystyle x(3x)(2x)=6$$ $$\displaystyle x^3=1$$ $$\displaystyle x=1\implies y=3,\,z=2$$ Observing that: $$\displaystyle C(1,3,2)=3(1)(2)+(3)(2)+2(1)(3)=18$$ and another constraint value such as$(x,y,z)=(1,2,3)\$ yields:

$$\displaystyle C(1,2,3)=3(1)(3)+(2)(3)+2(1)(2)=19$$

We may then conclude:

$$\displaystyle C_{\min}=C(1,3,2)=18$$