Minimizing the amount of material to make a cup

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In summary, an industrial is trying to minimize the amount of material used in making aluminum cups in the shape of a "V" shaped straight circular cylinder open at the top. The objective is to find the dimensions that use less material, with the constraint being the volume of the cup. After some discussion, the final objective function is S(r,h)=\pi r^2+2\pi rh, subject to the constraint V=\pi r^2h. The solution for R and H is (V/pi)^(1/3).
  • #1
leprofece
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An industrial makes a volume aluminum cups given "V" shaped straight circular cylinder open at the top. Find the dimensions that use less material.

Answer H = R = r = Cubic sqrt (3V/pi)

V = pir2h
And from here i don't know if it is right
r^2 = R2+ h2
as usual
I want to make sure
 
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  • #2
Re: max and min 7

leprofece said:
An industrial makes a volume aluminum cups given "V" shaped straight circular cylinder open at the top. Find the dimensions that use less material.

Answer H = R = r = Cubic sqrt (3V/pi)

V = pir2h
And from here i don't know if it is right
r^2 = R2+ h2
as usual
I want to make sure

Yes, the volume of a cylinder is given by:

\(\displaystyle V=\pi r^2h\)

However, you say "V"-shaped leading me to believe the cups are cones instead. You are trying to minimize the amount of material used, to what aspect of the cups will be our objective function?
 
  • #3
Re: max and min 7

to a part of it so i infer is the pitagorean one or maybe by thales I mean R/r = h-r/R
 
  • #4
Re: max and min 7

leprofece said:
to a part of it so i infer is the pitagorean one

Yes, the slant height of the cone is related to the radius and height of the cone via the Pythagorean theorem. Can you state the objective function and its constraint?
 
  • #5
Re: max and min 7

V = pir2h Objetive
And from here i don't know if it is right
r^2 = R2+ h2
as usual
I want to make sure Constraint so it must be
h2= r2-R2
 
  • #6
Re: max and min 7

I'm confused about how a cylinder can be "V"-shaped...I think the cups are conical, and so the objective function is:

\(\displaystyle S(r,h)=\pi r\sqrt{r^2+h^2}\) (the lateral surface area of a cone)

subject to:

\(\displaystyle V(r,h)=\frac{1}{3}\pi r^2h\) (the volume of a cone, which is a constant here)
 
  • #7
Re: max and min 7

MarkFL said:
I'm confused about how a cylinder can be "V"-shaped...I think the cups are conical, and so the objective function is:

\(\displaystyle S(r,h)=\pi r\sqrt{r^2+h^2}\) (the lateral surface area of a cone)

subject to:

\(\displaystyle V(r,h)=\frac{1}{3}\pi r^2h\) (the volume of a cone, which is a constant here)

An industrial makes a volume aluminum cups given "V" shaped straight circular cylinder open at the top. Find the dimensions that use less material.

oh my frind remenber i live in venezuela where people speaks spanish
so sorry it must be An industrial makes aluminum cups of a volume given "V" whose form is a straight circular cylinder shaped and open at the top. Find the dimensions that use less material.So it must be how you say
 
  • #8
Re: max and min 7

Okay, when you say "V" shaped, this to me sounds like a conical cup. But hey, we will get to the bottom of it yet! (Star)

So, what we are trying to minimize is the surface of a cylinder open at one end. Hence, our objective function is:

\(\displaystyle S(r,h)=\pi r^2+2\pi rh\)

Subject to the constraint:

\(\displaystyle V=\pi r^2h\implies h=\frac{V}{\pi r^2}\)

So, substitute for $h$ using the constraint, into the objective function, and you will have the surface area $S$ as a function of one variable, $r$, and then you may then minimize the function in the usual way.
 
  • #9
Re: max and min 7

I got R = H= (v/pi)1/3
 
  • #10
Re: max and min 7

leprofece said:
I got R = H= (v/pi)1/3

May you check please?
If The answer mine is right
 
  • #11
Re: max and min 7

leprofece said:
May you check please?
If The answer mine is right

I would be glad to check your work...otherwise I have to work the problem to see if your result is correct. :D
 

FAQ: Minimizing the amount of material to make a cup

How can we minimize the amount of material needed to make a cup?

One way to minimize the amount of material needed to make a cup is to use a thinner and lighter material. This could include materials such as paper, plastic, or even bamboo. Another method is to design the cup in a way that uses less material, such as a smaller volume or a more compact shape. Additionally, using recycled materials can also help reduce the overall amount of material needed.

Why is it important to minimize the amount of material used in making cups?

Minimizing the amount of material used in making cups is important for several reasons. First, it can help reduce the cost of production, making the cups more affordable for consumers. Second, it can also lead to a decrease in waste and a more sustainable use of resources. Lastly, using less material can also make the cups lighter and more portable, which can be beneficial for transportation purposes.

What are some challenges in minimizing the amount of material used in making cups?

One of the main challenges in minimizing the amount of material used in making cups is finding a balance between using enough material to make the cup durable and functional, while also using as little material as possible. Another challenge is finding materials that are both lightweight and environmentally friendly. Additionally, the cost of implementing new materials or designs can also be a challenge for companies.

Are there any potential drawbacks to minimizing the amount of material used in making cups?

One potential drawback of minimizing the amount of material used in making cups is that it may affect the overall quality and durability of the cup. Thinner materials may be more prone to breaking or cracking, and cups with less material may not be able to hold as much liquid. Additionally, using alternative materials or designs may also impact the overall aesthetic and appeal of the cup.

How can we ensure that minimizing the amount of material used in making cups is sustainable?

To ensure that minimizing the amount of material used in making cups is sustainable, it is important to consider the entire life cycle of the cup, from production to disposal. This includes using materials that are environmentally friendly and can be easily recycled or biodegraded. It is also important to assess the potential impact on the environment and to continually review and improve processes to reduce waste and minimize the use of resources.

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