Minimizing Material Used for a Box

In summary, the conversation is about someone seeking help with a problem set. They are specifically working on question two and are trying to find the correct dimensions for a box. The formula for surface area and volume are given, and the person is shown how to find the derivative of the surface area formula. The length they calculated is incorrect and they are asked to show their steps. The correct derivative is provided and they are asked to solve for the length. The person is unsure of what to do next.
  • #1
ardentmed
158
0
Hey guys,

I'm having trouble with this problem set I'm working on at the moment. I'd appreciate some help with this question:

(I'm only asking about number two, not one. Thanks.)
08b1167bae0c33982682_23.jpg


The lack of a top is accounted for in the formula for the box's surface area, which should be:

SA= L^2 +4HL
And the volume is simply:
32,000 = HL^2

THerefore, solving for the derivative of SA (or dSA) should give:

3L^2 - L^3 - 12,800.
L = 18.56635.

Ergo,
h=92.831.

Therefore, the dimension should be:
18.6cm x 92.8cm x 18.6cm

Am I close? Are the significant figures off?
Thanks in advance.
 
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  • #2
The length you got is wrong. Can you show me the your steps?

$$SA (l) = l^2 + \frac{4(32000)}{l}$$

Take its derivative and tell me what you get.
 
Last edited:
  • #3
Rido12 said:
The length you got is wrong. Can you show me the your steps?

$$SA (l) = l^2 + \frac{4(32000)}{l}$$

Take its derivative and tell me what you get.

I believe that the derivative is:

dSA = 2l - 128,000/(l^2)

Multiply the expression by the common denominator and solve for l.

Am I on the right track? What do I do from here?
 
  • #4
Yes, that is right. What do you get for the length?
 

Related to Minimizing Material Used for a Box

1. How can we reduce the amount of materials used for a box without compromising its durability?

There are several ways to minimize material used for a box without sacrificing its strength. One method is to use corrugated cardboard, which is made up of three layers and provides excellent strength and durability while using less material. Another way is to optimize the box design by using a structural analysis software, which can determine the minimum amount of material needed for a specific box size and weight capacity.

2. Can using recycled materials help in minimizing the amount of material used for a box?

Yes, using recycled materials is an effective way to reduce the amount of material used for a box. It not only minimizes the use of new raw materials but also reduces energy consumption and greenhouse gas emissions. However, it is important to ensure that the recycled materials meet the necessary strength and quality standards for the intended use.

3. Is there a specific box shape that uses less material compared to others?

Yes, there are box shapes that use less material than others. For example, a square or cuboid-shaped box uses less material than a rectangular or irregular-shaped box. This is because a square or cuboid provides more even distribution of stress, making it more efficient in terms of material usage.

4. How can we minimize the amount of material used for a box while keeping the packaging appealing to customers?

There are various ways to make a box visually appealing without using excess materials. One method is to use eco-friendly printing techniques, such as water-based inks, that use less material and have a lower environmental impact. Another way is to use creative and eye-catching designs that make use of negative space and do not require additional materials.

5. Are there any regulations or standards for minimizing material used for a box?

Yes, there are regulations and standards that aim to promote sustainable packaging and reduce the amount of material used for a box. For example, the Sustainable Packaging Coalition has developed guidelines for optimizing packaging design and reducing material usage. Additionally, many countries have regulations in place to encourage the use of eco-friendly materials and packaging practices.

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