Calc III Optimization problem (for dumpsters)

[Leinad7]

Homework Statement

"For this project we locate a trash dumpster in order to study its shape and constuction. We then attempt to determine the dimensions of a container of similar design that minimize construction cost."

1. (Already located, measured, and descibed a dumpster found).

2. "While maintaing the general shape and method of constuction, determine the dimensions such a container of the same volume should have in order to minimize the cost of construction. Use the following assumptions in your analysis:

* The sides, back, and front are to be made from 12-gauge (0.1046 inch thick) steel sheets, which cost $0.70 per square foot (including any required cuts or bends).

* The base is to be made from a 10-gauge (0.1345 inch thick) steel sheet, which costs $0.90 per square foot.

* Lids cost approximately $50.00 each, regardless of dimensions.

* Welding costs approximately $0.18 per square foot for material and labor combined.

- Give justification of any further assumptions or simplifications made of the details of construction.

3. Describe how any of your assumptions or simplifications may affect the actual result.

4. If you were hired as a constultant on this investigation, what would your conclusion be? Would you recommend altering the design of the dumpster? If so, describe the savings that would result."

Homework Equations

For this problem, going to choose a simple design of a dumpers, square. So we're going to have for the area equation v = base x height x width.

I'm not sure how I bring in the constraints due to costs into this problem.

The Attempt at a Solution

base=b, width=w, height=h. Therefore v=bwh. I think from this point I have to find the critical points, solving for them but not even sure how. I appreciate any advice on how to begin, or solved, thank you.

 

[Leinad7]

Sorry for not adding my work. Here it is so far:

V = b*w*h. Solving for h, h=v*(y*x)^-1

For my cost function c=.9*x*y+.7*2*x*v*(y*x)^-1+.7*2*y*v*(y*x)^-1+50
Simplifyed a bit more here,
c=.9*x*y+1.4v*y^-1+1.4v*x^-1+50
There's a problem with this, I have no included the welding costs associated with this, how might I add this in? Further more, once the welding cost is added, I take the derivative of the cost function to find local mins or maxs right? That seems to be the main problem at hand. I'm not sure how to deal with the x, and y unless I'm supposed to take partials derivatives and go from there.

 

[HallsofIvy]

You have to weld the edges where the faces meet. The cost of welding is $0.18 times the length of the edges. You don't, of course, weld the lid onto the frame!

 

[Leinad7]

Ah yes. You'd have to add those up. So in addition to the already existing cost function add the welding. Now do I take partials and find criticals?