Optimization - Find dimension of a cup that uses least amount of paper

Can you show your work?In summary, to find the height and radius of a cone-shaped paper drinking cup that holds 30 cm3 of water using the smallest amount of paper, you can use the equations for volume and surface area of a cone. By solving for the height and substituting it into the surface area equation, you can find the derivative of the surface area and set it equal to zero to solve for the radius. This will give you the optimal height and radius for the cup.
  • #1
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Homework Statement


A cone-shaped paper drinking cup is to be made to hold 30 cm3 of water. Find the height and radius of the cup that will use the smallest amount of paper.


Homework Equations


volume of a cone (1/3)(pi)(r^2)(h) = 30
SA of a cone pi(r)[sqrt(r^2 + h^2)]


The Attempt at a Solution


solve volume for h
plug h into SA
derive SA
calculate r
solve for h

I'm understanding how to do it, I just can't get the right answers.
 
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  • #2


Show your work, so we can know where did you go wrong.
 
  • #3


h = 90/(pir^2)

derivative of SA = 2pi*pir^2*r^2+90pi, denominator not needed, setting equal to zero.
 
  • #4


I got a different derivative (even after clearing out some denominators).
 

1. What is optimization and why is it important in finding the dimension of a cup that uses the least amount of paper?

Optimization is the process of finding the best solution for a problem within given constraints. In this case, it is important because we want to minimize the amount of paper used in making a cup, which can have environmental and cost implications.

2. What factors should be considered in optimizing the dimension of a cup to use the least amount of paper?

The factors that should be considered include the shape and size of the cup, the thickness of the paper, and the volume of the cup. These factors will affect the amount of paper needed to make the cup and how much space it takes up when stacked.

3. How do you determine the optimal dimension of a cup that uses the least amount of paper?

The optimal dimension can be determined through mathematical modeling and analysis, where various dimensions and factors are tested to find the most efficient combination. Computer simulations can also be used to speed up the process and find the optimal solution.

4. Is it possible to have a cup with zero paper wastage?

No, it is not possible to have a cup with zero paper wastage as some amount of paper will always be needed to make the cup. However, optimization can help minimize the amount of paper used, reducing wastage and improving efficiency.

5. Can optimization be applied to other products besides cups?

Yes, optimization can be applied to various products and processes in different industries. It can be used to minimize material usage, reduce costs, and improve efficiency in manufacturing and design. It is a valuable tool for problem-solving and improving sustainability.

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