optimization

[Dorian Gray]

Greetings Mathematicians,

I am having an extremely difficult time with this problem.

A rectangular storage container with an open top is to have a volume of 10 m^3. The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.

I have attached a copy of my work. I honestly do not know where to begin, and I do not understand the "length of its base is twice the width" in a drawing OR in the equation.

Any help is always greatly appreciated.


Here is a link to my work on flickr (It is larger and easier to see).

http://www.flickr.com/photos/77835031@N02/7004729537/#/

 

[tkhunny]

I am deeply troubled that you can produce all that and still declare "honestly do not know where to begin". This makes absolutely no sense at all.

Go slowly and logically. Dump out stuff you know and write it down in an organized fashion.

It's a rectangular storage container. It has three dimensions, length, width, and height. Let's call them L, W, and H respectively.

The volume of the box is L*W*H = 10 m^3 -- The formula is known and the 10 m^3 is given.

The area of the base is L*W. At a cost of $10 / m^2, this costs L*W*10 m^2

The area of the sides is 2*L*H + 2*W*H = 2*H*(L+W). At a cost of $6 / m^2, this costs 2*H*(L+W)*6 m^2

Total cost, then is L*W*10 m^2 + 2*H*(L+W)*6 m^2 = C(L,W,H)

This is where we have to take a step back. We need to minimize the total cost. How can we do that with all those variables in there? We must dispose of some of them. This is the reason for the additional information.

W know already that L*W*H = 10 m^3. Solve for H so we can get rid of it in the cost function.

H = 10 m^2 / L*W

C(L,W) = L*W*10 m^2 + 2*[10 m^2 / L*W]*(L+W)*6 m^2

Excellent. That is much better, but there are still two variables. How shall we remove another?

The Length is Twice the Width!! Just what we need. L = 2*W

This gives:

C(W) = (2*W)*W*10 m^2 + 2*[10 m^2 / (2*W)*W]*((2*W)+W)*6 m^2

After a little algebra, you should be good to go.

One step at a time. Never panic because you don't see the end. Keep making rational choices and you'll get there!

 

[Dorian Gray]

thank you

Hello JeffM,

Thank you very much for your response. Here is how my professor told us to do ALL optimization problems



Unfortunately, we did not even have the opportunity to go over the entire lesson before class was over, and it was the last period before break, which I am currently on at the moment. Furthermore, essentially all of our notes for optimization are not based on our book (they are specific notes with specific questions our instructor gave us), and our homework for this lesson is from the book. The guidelines above are the only way I know how to (attempt) to solve these.

 

[Dorian Gray]

thanks TKHunny

Thank you TKHunny for your very detailed response. These problems can be quite overwhelming (esp the first time you see and do them). I was also quite uncertain how to incorporate the costs into the function.

 

[Dorian Gray]

thank you

Thank you very much JeffM and TKhunny

@JeffM I have not heard of that Langrange's method yet, but I will certainly look at some videos online of it. Sounds interesting

@TKHunnny

Thank you for breaking it down into small pieces. I can clearly see that you must work from bits and pieces of information before you can actually set up your entire problem. I worked out the problem as you suggested, and I ended up with the correct answer! (yay!)

This "One step at a time. Never panic because you don't see the end. Keep making rational choices and you'll get there" was also very helpful was well! I think about that every time now that I address a problem.

Merci Boucoup (Thank you very much)
DG