Optimizing Dimensions to minimize Cost

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In summary, to determine which dimensions to optimize for cost reduction, you must identify key factors affecting cost and use data analysis and modeling techniques. Methods such as linear programming, simulation, and sensitivity analysis can be used to optimize dimensions. Optimizing dimensions can have a positive or negative effect on product quality, and challenges may include finding the right balance between cost and quality and the complexity of data analysis. The frequency of re-evaluating dimensions for cost optimization will vary depending on the product and industry, but regular reviews should be conducted to identify potential areas for optimization.
  • #1
Rayquesto
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Homework Statement



A metal can which holds 450 cm^3 is to be manufactured in cylindrical shape. The top and bottom will be cut from two squares and the corner scrap discarded (but paid for). The metal for the sides costs 0.1 cents/cm^2, while the cost for the top and bottom is 0.15cents/cm^2. Allow 0.5cents/cm for the vertical or side seam and 0.6 cents/cm for the seams joining top and bottom to the sides.
a) Determine the dimensions that would yield a minimum cost. Give dimensions to the nearest thousandth of a centimeter.
b) Determine the corresponding cost to the nearest hundreth of a cost.

Homework Equations



Minimizing is to find the values of a unit of function, when the tangent slope is zero.

therefore,

If C'(A)=0, then we have to find the dimensions when this happens since this is when cost is minimized.

V(cylinder)=∏r^2h
A(cylinder)=2∏r(h) + 2∏r^2

The Attempt at a Solution



I notice that part of the problem states that no matter what dimension, V=450cm^3.

Therefore,

since this is a can cylinder,

V=∏r^2h=450cm^3

and so it is useful to note that:

(450cm^3)/(∏r^2)=h for all h>0 and r>0
and
(√((450cm^3)/(h∏)))=r for all h>0 and r>0

So, here's how I went along trying to put certain pieces together:

According to my interpretation of the problem,

Area(top + bottom)=2(∏r^2)
So,
Cost(Area(top + bottom))=[.15cents/cm^2][2∏r^2]

Area(side)=2∏r(h) since my intuition tells me this represents the area of a cross section of the side of a cylinder where 2∏r is the circumference of the number of circles you multiply by to get the area

Cost(Area(side))= .15cents/cm^2

So also since for every radius there is a cost and every height there is a cost (I am very currently skeptical about this intuition!)

Cost(Area(side + top + bottom))= .3∏r^2[Cost(r)]^2 + .2∏r(h)[Cost(r)][Cost(h)]

Therefore, by substituting h=(450cm^3)/(∏r^2)

Cost(Area(side + top + bottom))= .3∏(r)^2[cost(r)] + .2∏r(450cm^3/∏r^2)[cost(r)][cost(h)]

Cost(h)=(.6cent/cm)h=(.6cent/cm)[(450cm^3)/(∏r^2)]
cost(r)=(.5cent/cm)r

Cost(Area(side + top + bottom))=.3∏r^2[(.5cent/cm)r] + .2∏r[450cm^3/∏r^2][(.5cent/cm)r]

Cost'(Area(side + top + bottom))=.9∏[.5cent/cm]r^3 + 45=0 to minimize cost, but r>0 and h>0 what am I doing wrong?
 
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  • #2
Rayquesto said:

Homework Statement



A metal can which holds 450 cm^3 is to be manufactured in cylindrical shape. The top and bottom will be cut from two squares and the corner scrap discarded (but paid for). The metal for the sides costs 0.1 cents/cm^2, while the cost for the top and bottom is 0.15cents/cm^2. Allow 0.5cents/cm for the vertical or side seam and 0.6 cents/cm for the seams joining top and bottom to the sides.
a) Determine the dimensions that would yield a minimum cost. Give dimensions to the nearest thousandth of a centimeter.
b) Determine the corresponding cost to the nearest hundreth of a cost.

Homework Equations



Minimizing is to find the values of a unit of function, when the tangent slope is zero.

therefore,

If C'(A)=0, then we have to find the dimensions when this happens since this is when cost is minimized.

V(cylinder)=∏r^2h
A(cylinder)=2∏r(h) + 2∏r^2

The Attempt at a Solution



I notice that part of the problem states that no matter what dimension, V=450cm^3.

Therefore,

since this is a can cylinder,

V=∏r^2h=450cm^3

and so it is useful to note that:

(450cm^3)/(∏r^2)=h for all h>0 and r>0
and
(√((450cm^3)/(h∏)))=r for all h>0 and r>0

So, here's how I went along trying to put certain pieces together:

According to my interpretation of the problem,

Area(top + bottom)=2(∏r^2)
So,
Cost(Area(top + bottom))=[.15cents/cm^2][2∏r^2]

Area(side)=2∏r(h) since my intuition tells me this represents the area of a cross section of the side of a cylinder where 2∏r is the circumference of the number of circles you multiply by to get the area

Cost(Area(side))= .15cents/cm^2
You said above, "The metal for the sides costs 0.1 cents/cm^2"

So also since for every radius there is a cost and every height there is a cost (I am very currently skeptical about this intuition!)

Cost(Area(side + top + bottom))= .3∏r^2[Cost(r)]^2 + .2∏r(h)[Cost(r)][Cost(h)]

Therefore, by substituting h=(450cm^3)/(∏r^2)

Cost(Area(side + top + bottom))= .3∏(r)^2[cost(r)] + .2∏r(450cm^3/∏r^2)[cost(r)][cost(h)]
You also need to include the cost of the seams- "Allow 0.5cents/cm for the vertical or side seam and 0.6 cents/cm for the seams joining top and bottom to the sides." The vertical seam has length h so you need to add 0.5h cents and the top and bottom seams have length [itex]2\pi r[/itex], and there are two of them, you need to add [itex](0.5)(4\pi r)= 2\pi r[/itex].

Cost(h)=(.6cent/cm)h=(.6cent/cm)[(450cm^3)/(∏r^2)]
cost(r)=(.5cent/cm)r

Cost(Area(side + top + bottom))=.3∏r^2[(.5cent/cm)r] + .2∏r[450cm^3/∏r^2][(.5cent/cm)r]

Cost'(Area(side + top + bottom))=.9∏[.5cent/cm]r^3 + 45=0 to minimize cost, but r>0 and h>0 what am I doing wrong?
 
  • #3
The cost of the top + bottom is (0.15)*2*(2r)^2, because you pay 0.15 per cm^2 for the whole 2r by 2r square (you pay for the discarded corners, you said).

RGV
 

FAQ: Optimizing Dimensions to minimize Cost

1. How do you determine which dimensions to optimize in order to minimize cost?

In order to determine which dimensions to optimize, you must first identify the key factors that affect cost in your specific situation. This could include materials, production processes, transportation, and packaging. Once these factors are identified, you can use data analysis and modeling techniques to determine which dimensions have the greatest impact on cost and should be prioritized for optimization.

2. What methods can be used to optimize dimensions for cost reduction?

There are a variety of methods that can be used to optimize dimensions for cost reduction. These include mathematical optimization techniques such as linear programming, simulation and modeling, sensitivity analysis, and design of experiments. These methods can help identify the ideal combination of dimensions that will result in the lowest cost.

3. How can optimizing dimensions affect product quality?

Optimizing dimensions for cost reduction can have both positive and negative effects on product quality. On one hand, reducing costs can lead to lower quality materials or processes, resulting in a lower quality product. However, if done carefully and with consideration to product specifications, optimizing dimensions can also result in cost savings without sacrificing quality.

4. What are some potential challenges in optimizing dimensions for cost reduction?

One potential challenge in optimizing dimensions for cost reduction is finding the right balance between cost and quality. It's important to consider the potential trade-offs and unintended consequences of cost reduction in terms of product performance and customer satisfaction. Additionally, data analysis and modeling can be complex and require specialized expertise and resources.

5. How often should dimensions be re-evaluated for cost optimization?

The frequency of re-evaluating dimensions for cost optimization will depend on the specific product and industry. In general, it's good practice to regularly review and analyze cost data to identify any areas for potential optimization. This could be done on a quarterly, bi-annual, or annual basis, depending on the level of cost savings and changes in the market or production processes.

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