Optimizing surface area of a silo

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Homework Help Overview

The discussion revolves around optimizing the surface area and cost of construction for a silo, which consists of a cylindrical wall, a flat circular floor, and a hemispherical top. The original poster is comparing two problems: one focusing on minimizing surface area for a given volume and the other on minimizing construction costs, which are affected by the different costs associated with the hemispherical top.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • The original poster attempts to adapt their previous solution for minimizing surface area to the new problem of minimizing cost, questioning what changes are necessary in their approach.
  • Some participants suggest focusing on the cost function, which includes the areas of the silo components with different cost factors.
  • There are inquiries about the correct interpretation of the ratio of height to diameter in the context of the problem.
  • Participants express confusion regarding the implications of the hemispherical top's cost on the overall height calculation.

Discussion Status

Participants are actively engaging with the problem, offering clarifications about the cost structure and discussing the necessary adjustments to the original approach. There is a lack of explicit consensus on the correct formulation of the ratio of height to diameter, indicating ongoing exploration of the problem.

Contextual Notes

There are constraints related to the differing costs of construction for the hemispherical top compared to the cylindrical wall and floor, which are central to the problem. The original poster is also under time pressure due to an impending homework deadline.

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A silo has cylindrical wall, a flat circular floor, and a hemispherical top. If the cost of construction per square foot is twice as great for the hemispherical top as for the walls and the floor, find the ratio of the total height to the diameter of the base that minimizes the total cost of construction.

There was another problem involving the silo which said: for a given volume, find the ratio of the total height to the diameter of the base that minimizes the total surface.

For this I wrote the equation for volume and solved for h and then I plugged that into the equation for area of the silo, derived, set equal to zero and found that radius and height both equal cubed route of 3v/5pi which would make the ratio: h+r/2r = 1.

What do I have to do differently for this question? Thanks.
 
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any advice on this? I'm really stuck and this hmwk is due tomorrow.
 
The cost is proportional to twice the area of the top plus the area of the other parts. That's the only thing you have to change. Minimize cost instead of area.
 
Dick said:
The cost is proportional to twice the area of the top plus the area of the other parts. That's the only thing you have to change. Minimize cost instead of area.

So I'm just minimizing A = (pi)R^2 + 2(pi)RH + 4(pi)r^2?
 
Yes.
 
Thx brotha
 
So I got R and H = (3V/11pi)^(1/3). Would the ratio be 2H + R / 2R or 3/2?
 
"So I got R and H = (3V/11pi)^(1/3)"

What does that mean? You got what for R? The "ratio of height to diameter of base" is H/(2R).
 
well it says total height so the height of the cylinder + the hemisphere which would be
H + R/(2R). I thought that since the hemisphere costs twice as much it would be 2H. And I got R = H = (3V/11pi)^(1/3). So where'd I mess up? Thanks.
 

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