Minimizing Surface Area/Volume

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SUMMARY

The discussion focuses on optimizing the design of a capsule composed of two hemispheres and a cylindrical section to minimize material costs while maintaining a volume of 0.25 cubic meters. The cost of materials is $0.0025 per square centimeter for one hemisphere and $0.0015 per square centimeter for the other hemisphere and the cylinder. Key equations include the volume of the sphere (V = 4/3πr³) and the surface area of the cylinder (SA = 2πr² + 2πrh). The goal is to determine the optimal dimensions that minimize total material costs.

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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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