Minimizing Surface Area/Volume

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In summary, the problem is to find the optimal measurements for a large capsule (with hemispherical top and bottom and cylindrical middle) that can hold 0.25 cubic meters of medicine. The materials cost for the top hemisphere is $0.0025 per square centimeter, while the bottom hemisphere and cylinder have a cost of $0.0015 per square centimeter. The relevant equations for calculating volume and surface area of a sphere and cylinder are provided. The solution involves finding the dimensions that minimize the total materials cost for the capsule and calculating the corresponding cost. The poster is asked to show their attempt and indicate where they are having difficulty.
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Mldancer125
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Homework Statement


I need to find a solution to make a large capsule ( top and bottom are hemispheres and middle is a cyliner) The capsule must be big enough to hold .25 cubic meters of medicine. One hemisphere's materials costs $.0025 per square centimeter and the other hemisphere and cylinder materials costs $ .0015 per square centimeter. I need to know the optimal measurements to minimize the total materials cost for the case, as well as the total materials cost for this optimal design.


Homework Equations



V of sphere= 4/3pi r^3
V of cylinder= pi r^2h
SA of sphere=4 pi r^2
SA of cylinder= 2pi r^2+ 2pi rh

The Attempt at a Solution

 
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Welcome to PF!

Hi Mldancer125! Welcome to PF! :wink:

Show us what you've tried, and where you're stuck, and then we'll know how to help! :smile:
 

1. What is the purpose of minimizing surface area/volume?

The purpose of minimizing surface area/volume is to reduce the amount of material or energy required to create or sustain a structure or object. This can lead to cost savings, increased efficiency, and reduced environmental impact.

2. How does minimizing surface area/volume impact the strength of a structure?

Minimizing surface area/volume can actually increase the strength of a structure. This is because reducing the amount of material used can create a more compact and efficient design, resulting in a stronger structure that can better withstand external forces.

3. Are there any limitations to minimizing surface area/volume?

Yes, there are limitations to minimizing surface area/volume. While it can lead to cost savings and increased efficiency, it may also result in a decrease in functionality or practicality. For example, a structure with minimal surface area may not have enough space for necessary components or may not be able to support heavy loads.

4. How does minimizing surface area/volume impact sustainability?

Minimizing surface area/volume can have a positive impact on sustainability. By reducing the amount of material used, there is less waste and environmental impact in the production and maintenance of a structure or object. It can also lead to energy savings in the long run.

5. What are some common applications of minimizing surface area/volume in science and technology?

Minimizing surface area/volume is commonly used in various fields such as architecture, engineering, and manufacturing. It is also important in biology, as organisms often have efficient designs that minimize surface area/volume to aid in functions such as respiration and movement. In technology, this concept is applied in the design of compact and energy-efficient devices such as smartphones and laptops.

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