# Cost minimization while retaining volume

#### mikezap99

I need to design a dumpster for my final calculus project. The main idea is that I find a dumpster, measure it, and retain the basic shape and construction when redesigning it. I must also keep the volume constant.

The problem would be a heck of a lot easier if it was just a rectangular prism, but it's a trapezoidal prism. The cost for the sides, front, and back is $0.70 cents per square foot (including any cuts or folds). The base costs$0.90 per square foot. The lid costs $50.00 regardless of dimensions. Welding, for joints, costs$0.18 per foot. This would be for side lengths, etc.

I know that I will have to use partial differentiation, but I'm at a loss as to how to start this and the general idea with the process of this problem. I can't use a rectangular prism instead of the trapezoid. Any help or explanations are greatly appreciated.

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#### TKHunny

1) "Retain the basic shape" might include squaring it up just a bit. Is this specifically proscribed?

2) The total volume is not 3.846. It's more than that, but less than 3.912. Knowing where to start might be the first hurdle to overcome.

3) Thought question: Are you sure you have to work with the whole thing at the same time. Will you get the same answer if you put 3D coordinate axes in the middle and work with the first octant? If that will work, and you need to decide if it will, that might square things up a bit for the arithmetic.

#### mikezap99

The volume is definitely 3.846 meters cubed. I changed the dimensions into feet and the total volume is 135.762 feet cubed. The shape must be a trapezoidal prism... I've made some headway.

The variables x,y,z, and g represent the width, length, back height, and front height, respectively.

Volume = (1/2)(g+z)xy
And total cost =
0.7(xg+xz+yz+yg)+0.9xy+0.36(x+y+z+g)

Since volume = 135.762,
g and z can be eliminated
So, cost= 190.0668/y + 190.0668/x +0.9xy+0.36x+0.36y+162.9144/(xy)

f(x,y)=190.0668/y+190.0668/x+0.9xy+0.36x+0.36y+162.9144/(xy)

Take the first derivatives with respect to each x and y
fx(x,y)= -190.0668/x2 + 0.9y + 0.36 - 162.9144/((x^2)y)

fy(x,y)= -190.0668/y2 + 0.9x + 0.36 - 162.9144/(x(y^2))

Where do I go from here? How do I use what I have to find the minimums for the cost calculation?

#### TKHunny

It's still not 3.846. It is if 1.500 = 1.475. Either that or you measured it incorrectly and it's not 1.500 where it says it is.

$$\displaystyle \frac{1}{2} \cdot (1.565+1.285) \cdot 1.830 \cdot 1.475 = 3.846$$

$$\displaystyle \frac{1}{2} \cdot (1.565+1.285) \cdot 1.830 \cdot 1.500 = 3.912$$

You may not just pick one. The right answer is in there, somewhere.

#### mikezap99

The 1.5 m is along the slanted, top section of the trapezoidal prism and isn't necessary for the calculation. My group member and I measured it and included it in our preliminary diagrams in case it ended up being useful. The correct volume is 3.846 meters cubed. Sorry for being unclear when the question was posted originally.