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burhan101
 2 years ago
A cylinder can is to have a volume of 900 cm cubed. The metal costs $15.50/squared meter. What dimensions produce a can with minimum cost? What is the cost of making the can?
burhan101
 2 years ago
A cylinder can is to have a volume of 900 cm cubed. The metal costs $15.50/squared meter. What dimensions produce a can with minimum cost? What is the cost of making the can?

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e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1I see an issue with your derivative.

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1\(\cfrac{27900}{r}=27900r^{1}\) Use this second form of it and the exponent rules you know and see what you get.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0i get dw:1371523952498:dw

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Yes. Your other part, the \(+62\pi r\) is fine, so now add those fractions up and see where you go.

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Your mistake was in how you took the derivative of that. Other than that, the next steps looked pretty good, but got fowled up by involving the wrong C'.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0Was the c' was wrong due to my wrong derivative?

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1You come up with a new, reduced equation? In your first one, you elimitated the \(r^2\) on the bottom. In this, there is a little more you can cancel out. The 62.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1371526270946:dw

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0the r^2 cancel out. what else

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Well, it does not really cancel... it is because there is a 0 on the other side that it does not matter.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0\[\huge 0=62\pi r27900 \]

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0and then I just solve for 'r' ?

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Now, what if you multiply through by 1/62?

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0cant I do dw:1371526627893:dw

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Ah, in the earlier one too. You can't cancel the part of the \(r^3\) above. Like I said, it is not that it cancels but that it does not matter.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0ohhh okay let me fix that

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1\(62\pi r^3  27900=0\)

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0theres a denominator too right ^

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Because if the top of the fraction is 0, it is 0. Only if the bottom woulc cause an asymptote does it matter....

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0cant i just multiply the equation by r^2 to get rid of the denominator

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0oh okay, so i only focus on the top

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0So like mathematically on my paper that my prof would mark i would just ignore it, like cant i be docked marks ?

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Yah. Mathematically, it is the same as multiplying through by \(r^2\) because the right hand side is 0 so \(0\times r^2=0\) means it does not change.

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1I have the entire reducing this fraction in long form with every step. Just finished writing it up. So I'll post it when we get there and you can check what you have against it.

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1dw:1371527305050:dw

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0Okay thanks for that, ill check it as soon as i finish !! :D

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Well, solve for the root, and tell me what you get. Then I'll post what I got, what I did, and even a graph that shows some things.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0would I leave the calculator in radians?

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Does not matter. No degrees involved.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1371528718316:dw

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1pi is a constant, not an angle. So yah. \[\cfrac{62\pi r^3  27900}{r^2}=0\implies \\ \\ r^2\times\cfrac{62\pi r^3  27900}{r^2}=r^2\times0\implies \\ \\ 62\pi r^3  27900=0\implies \\ \\ \cfrac{1}{62}\times(62\pi r^3  27900)=\cfrac{1}{62}\times0\implies \\ \\ \pi r^3  450=0\implies \\ \\ \pi r^3  450+450=0+450\implies \\ \\ \pi r^3 =450\implies \\ \\ \cfrac{1}{\pi}\pi r^3 =\cfrac{1}{\pi}450\implies \\ \\ r^3 =\cfrac{450}{\pi}\implies \\ \\ \sqrt[3]{ r^3} =\sqrt[3]{ \cfrac{450}{\pi}}\implies \\ \\ r =\sqrt[3]{ \cfrac{450}{\pi}} \]And in the graph you can see how what I start with and end with both overlap, and they are 0 right where you said, about 5.23. https://www.desmos.com/calculator/mz0fk3daat

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0Oh my, thank you SO much !

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0now can I plug this r value intto my cost equation ?

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Here is another interesting point that is very good to know for these types of problems. Let's take the test points of 5 and 6 and put them into \(\pi r^3  450\) \(\pi (5)^3  450\approx 57\) \(\pi (6)^3  450\approx 229\) So as x is increasing, this is moving from negative, to zero, then positive. That means the original equation has a slope there that is negative, bottoms out, then goes positive. This confirms that what you found is a minimum!

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Yes, you can put that r into the original.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0ohhh, i can pick any two numbers?

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Well, close numbers are best for this sort of test. They just become test points on each side of the critical point to confirm if it is a min or max. If you cross over two critical points, that sort of test is invalid. but we only have one critical point, so it is no big deal.

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1I do see one big thing to be careful of in all of this.

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1"A cylinder can is to have a volume of 900 cm cubed." \(\leftarrow\) in cm. "The metal costs $15.50/squared meter." \(\leftarrow\) in m! Watch out for your units!

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0thanks for the heads up !

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Oops. made a mistake there... I put cm cubed, but it is squared. Because 1m = 100 cm, so square both and you get: \(1m^2=10000cm^2\) The cubic relationship here is: \(900 cm^3 = .0009 m^3\) So you need to be careful because you found the radius in cm. So your can will be in cm.

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0I know, i still understood what you were saying :P

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0\[\large C=7998.49m^2\]

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1\(\cfrac{$15.50}{10000cm^2}\times 5.23cm^2\) The \(cm^2\) cancels....

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1So like $0.0081 per can.... less than a penny each. Make sure you got the units right in the original, but if so, this is like the bulk manufacture of soda can...

burhan101
 2 years ago
Best ResponseYou've already chosen the best response.0back of the book says $0.80 each

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Hmmm... I must have converted wrong somewhere.

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1I don't see where though... odd.... because that would be like it was a linear conversion and this is a square conversion.

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Ah! Remembered. R needs to go back into the formula!

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1\(\cfrac{1800}{r}+2\pi r^2\implies \cfrac{1800}{5.23}+2\pi (5.23)^2=????\) That makes much more sense!

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Yes, that got me something that will round to 80 cents.

e.mccormick
 2 years ago
Best ResponseYou've already chosen the best response.1Yah, it also explains where I made a mistake. I put it back into the wrong equation! I took the linear radius when I needed the square surface area! Be very careful of that.
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