Pine costs twice as much as fibreboard

  • Thread starter danizh
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In summary, Sindy is trying to find the dimensions of a wooden box that will cost the least amount of money to build. She has decided to use fiberboard for the back and bottom, and pine for the top, front, and two sides. The dimensions of the box are 0.516 x 1.032 x 0.563.
  • #1
danizh
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I've been stuck on this question for more than an hour and I still can't seem to get the right answer (:mad::mad:). So here goes:

Question:Sindy is building a rectangular wooden btoy chest so that its length is two times its width. She has picked pine for the top, front, and two sides of the chest. She will use fibreboard for the back and bottom. The chest much have volume of 0.3m^3. Pine costs twice as much as fibreboard. Find the dimensions that will minimize the cost of the chest.

My wrong calculations:
Let a be the price of fibreboard, and 2a be price of pine.
2x^2y = 0.3
so 7 = 0.3/(2x^2)
Cost = 2a (2xy + 2x^2 + 2xy) + a (2xy+2x^2)
= 10axy + 6ax^2
= 10ax(0.3/2x^2) + 6ax^2
= (1.5a/x) + 6ax^2

Now if I take the derivative of this, I get:

Cost' = -1.5x^(-2) + 12x
1.5 = 12x^3
0.125 = x^3
x = 0.5

And so, y=0.6

The correct answer shold be 0.585m x 1.170m x 0.438m, so any help would be great as to what I'm doing wrong!
 
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  • #2
danizh said:
I've been stuck on this question for more than an hour and I still can't seem to get the right answer (:mad::mad:). So here goes:
Question:Sindy is building a rectangular wooden btoy chest so that its length is two times its width. She has picked pine for the top, front, and two sides of the chest. She will use fibreboard for the back and bottom. The chest much have volume of 0.3m^3. Pine costs twice as much as fibreboard. Find the dimensions that will minimize the cost of the chest.
My wrong calculations:
Let a be the price of fibreboard, and 2a be price of pine.
2x^2y = 0.3
so 7 = 0.3/(2x^2)
Cost = 2a (2xy + 2x^2 + 2xy) + a (2xy+2x^2)
= 10axy + 6ax^2
= 10ax(0.3/2x^2) + 6ax^2
= (1.5a/x) + 6ax^2
I guess x is the width, so 2x is the length and y is the height of the box. Still, it'd be helpful if you stated what the variables you introduce mean and you should give them symbols that are 'obvious' (i.e. l,w,h for length width, height) for several reasons:
(1) Your work becomes easier to follow so other can help you more easily.
(2) It becomes easier to check you work yourself
(3) It will prevent confusion, which I think is the reason it went wrong here.

The area of pine is: 2lh+wh+lw (2 sides + front + top)
the area of board is: lw+wh (bottom + back)
So the cost is: 2a(2lh+wh+lw)+a(lw+wh). Or, since l=2w:
Cost = 2a(5wh+2w^2)+a(2w^2+wh)
Which is already different from your result:
Cost = 2a (4xy + 2x^2) + a (2x^2+2xy)

EDIT: Wait a sec. I don't have a picture of the box. Is the area of the back of the box length times height or width times height? I assumed it was wh, but if it's lh your expression is correct.
 
Last edited:
  • #3
It isn't specified, that is what's causing the confusion. In a situation like this, would you consider "width x height" the back (the shorter length multiplied by height)?
 
  • #4
The question is pretty vague on that. I thought it was obvious the length (longer side) would correspond to a side and the width to the back.
Anyway, I get that the dimensions are 0.516 x 1.032 x 0.563.
So I guess the answer in your book is wrong. w=0.516 is definitely the value which minimizes 2(pine area)+(board area) under the given constraints.
 
  • #5
Makes sense, thanks a lot Galileo! :)
 

FAQ: Pine costs twice as much as fibreboard

What is the reason for the price difference between pine and fibreboard?

The main reason for the price difference between pine and fibreboard is the materials used for their production. Pine is a natural wood that requires harvesting, processing, and transportation, while fibreboard is made from recycled wood fibers and can be produced in a factory setting.

Does the quality of pine justify its higher cost compared to fibreboard?

The quality of pine is generally considered to be superior to fibreboard due to its durability, strength, and natural beauty. However, the decision to use pine or fibreboard ultimately depends on the specific needs and budget of the project.

Are there any other factors that contribute to the higher cost of pine?

In addition to the materials used, other factors that may contribute to the higher cost of pine include the demand for this type of wood, transportation costs, and the location of the supplier. These can vary depending on the market and availability of pine in different regions.

Is it worth paying twice as much for pine over fibreboard?

Whether or not it is worth paying twice as much for pine over fibreboard depends on the individual's specific needs and preferences. For some projects, the natural qualities of pine may be worth the higher cost, while for others, the cost savings of using fibreboard may be more appealing.

What are the alternatives to using pine or fibreboard for a project?

There are various alternatives to using pine or fibreboard for a project, including other types of wood such as cedar or oak, as well as synthetic materials like laminate or PVC. The best alternative will depend on the specific requirements and budget of the project.

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