# Package Optimization problem

1. Jul 22, 2004

### decibel

A choclate manufacturer uses an equilateral trianglular prism package. if the volume of chocklate to be contained in the package is 400 cm ^3 . what dimenesions of the package will use the minumum amount of materials?

i'm having trouble putting the formulas together, im thinking of the front and back of the prism as being A=2(1/2bh) then i dont know wut to do with the 3 rectangular peices in the middle....can someone push me in the right lane here?...this is urgent, i need to do this in the next couple of hours, thanks in advance.

2. Jul 22, 2004

### Jin314159

I understand that this is a triangular prism.

Since the triangular is equilateral, this triangular prism is defined by only two parameters: the length of the side of the triangle (call it L) and the length of the sides connecting the two triangles (call it M).

Since the volume of the triangular prism has to equal 400. We get one equation, namely the volume as a function of L and M, Volume(L,M) = 400. That is our restraint equation.

Now we want to minimize the amount of material, which I interpret as pretty much the same as the surface area (unless someone else thinks of something else better). So we take the surface area as a function of L and M, SurfaceArea(L,M). We plug in stuff we know from the restraint to get L in terms of M or vice versa. Then we will get surface area as a function of only one variable SurfaceArea(L) or SurfaceArea(M). Then minimize the function with respect to the variable. Then use the restraint equation to figure out the other variable.

Hope all this makes sense.

3. Jul 22, 2004

### HallsofIvy

Staff Emeritus
You are starting out correctly: the two ends each have area (1/2)bh so the total area of the two ends will have area bh. But what is h? Since you are told that this is an "equlateral triangular prism", if you divide the end triangle into two right triangles, each has hypotenuse of length b, one leg of length (1/2)b and one leg of h. Now use the Pythagorean formula to find h in terms of bl: h= (sqrt(3)/2)b. Of course, each rectangular side has area bw where w is the length of the package. The total area is
sqrt(3)b+ 3bw while the volume is (1/2)bhw= sqrt(3)b2w= 400. Since we are not given any relation between w and b, we will need to treat this as a function of two independent varialbles and use the "Laplace multiplier" method. The gradient of the area function is the vector <sqrt(3)+ 3w, b> while the gradient of the condition is <2sqrt(3)bw, &radic;(3)b2>. Since one must be a multiple of the other, we must have 2√(3)bw= &lambda;(√(3)+ 3w) and sqrt(3)b2= &lambda; b.

Last edited: Jul 23, 2004
4. Jul 27, 2004

### decibel

help appreciated thanks