Most efficient cost for a cylinder

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Homework Help Overview

The problem involves finding the dimensions of an open-topped cylinder that will minimize the total cost of materials while maintaining a specified volume of 250 cm³. The costs for the materials of the bottom and the side of the cylinder are given, which adds a layer of complexity to the optimization task.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to set up equations for the areas and costs associated with the cylinder's bottom and side. They express the total cost in terms of the radius and height, and begin to differentiate to find a minimum cost. Some participants question the formatting of the mathematical expressions and the clarity of the original post.

Discussion Status

The discussion has seen the original poster working through the problem and expressing some confusion regarding formatting and errors in their calculations. They have indicated a resolution to their initial formatting issues but have not provided a final solution or conclusion to the problem. There is no explicit consensus on the approach yet, as the focus has been on clarifying the setup and addressing errors.

Contextual Notes

Participants have noted the importance of maintaining clarity in mathematical expressions, which may affect the understanding of the problem. The discussion reflects a learning environment where participants are encouraged to explore their reasoning and clarify their thoughts without rushing to conclusions.

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Homework Statement


An open-topped cylinder is to have a volume of 250 cm3. The material for the bottom of the pot costs 4 cents per cm2, and the material for the side of the pot costs 2 cents per cm2. What dimensions will minimize the total cost of this pot?

The Attempt at a Solution


$$
A_{bottom}=πr^2
\\
C_{bottom}=4(πr^2)
$$

$$
A_{side}=2πrh
\\
C_{side}=2(2πrh)
$$

$$
V=πr^2h
\\
250=πr^2h
\\
h=\frac {250}{πr^2}
\\
∴C_{side}=2(2πr\frac {250}{πr^2})
$$

$$
C_{total}=4(πr^2+2(2πr\frac {250}{πr^2})
\\
\frac {d(C_{total})}{d(r)}=8πr-\frac{1000}{πr^3}
$$

Then I tried to use the first derivative test. I am stuck.
 
Last edited:
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Uhh I messed something up there with itex.
 
I really messed up the formatting in that first post so it kind of looks like a mess. Until I figure that out, perhaps someone could point me in the right direction to solving the question?
 
Okay I think I fixed all the formatting. Silly me.
 
NEVERMIND. I figured out my silly error. It's all good now. Can I delete this thread?
 
EricPowell said:
NEVERMIND. I figured out my silly error. It's all good now. Can I delete this thread?
We don't delete threads as a matter of course. Even though it's of no use to you now, others might find it helpful.
 

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