anonymous
  • anonymous
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?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
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anonymous
  • anonymous
1 Attachment
anonymous
  • anonymous
Am I doing it right?
e.mccormick
  • e.mccormick
I see an issue with your derivative.

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anonymous
  • anonymous
please point it out
e.mccormick
  • e.mccormick
\(\cfrac{27900}{r}=27900r^{-1}\) Use this second form of it and the exponent rules you know and see what you get.
anonymous
  • anonymous
i get |dw:1371523952498:dw|
e.mccormick
  • e.mccormick
Yes. Your other part, the \(+62\pi r\) is fine, so now add those fractions up and see where you go.
e.mccormick
  • e.mccormick
Your 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'.
anonymous
  • anonymous
Was the c' was wrong due to my wrong derivative?
e.mccormick
  • e.mccormick
Yes.
e.mccormick
  • e.mccormick
You 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.
anonymous
  • anonymous
|dw:1371526270946:dw|
anonymous
  • anonymous
the r^2 cancel out. what else
e.mccormick
  • e.mccormick
Well, it does not really cancel... it is because there is a 0 on the other side that it does not matter.
anonymous
  • anonymous
\[\huge 0=62\pi r-27900 \]
anonymous
  • anonymous
and then I just solve for 'r' ?
e.mccormick
  • e.mccormick
Now, what if you multiply through by 1/62?
anonymous
  • anonymous
cant I do |dw:1371526627893:dw|
e.mccormick
  • e.mccormick
Forgot your cube.
anonymous
  • anonymous
what?
e.mccormick
  • e.mccormick
Ah, 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.
anonymous
  • anonymous
ohhh okay let me fix that
e.mccormick
  • e.mccormick
\(62\pi r^3 - 27900=0\)
anonymous
  • anonymous
theres a denominator too right ^
e.mccormick
  • e.mccormick
Because if the top of the fraction is 0, it is 0. Only if the bottom woulc cause an asymptote does it matter....
anonymous
  • anonymous
cant i just multiply the equation by r^2 to get rid of the denominator
anonymous
  • anonymous
oh okay, so i only focus on the top
anonymous
  • anonymous
So like mathematically on my paper that my prof would mark i would just ignore it, like cant i be docked marks ?
e.mccormick
  • e.mccormick
Yah. 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
  • e.mccormick
I 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
  • e.mccormick
|dw:1371527305050:dw|
anonymous
  • anonymous
Okay thanks for that, ill check it as soon as i finish !! :D
e.mccormick
  • e.mccormick
Well, 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.
anonymous
  • anonymous
would I leave the calculator in radians?
e.mccormick
  • e.mccormick
Does not matter. No degrees involved.
anonymous
  • anonymous
the pi ?
anonymous
  • anonymous
|dw:1371528718316:dw|
e.mccormick
  • e.mccormick
pi 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
anonymous
  • anonymous
Oh my, thank you SO much !
anonymous
  • anonymous
now can I plug this r value intto my cost equation ?
e.mccormick
  • e.mccormick
Here 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
  • e.mccormick
Yes, you can put that r into the original.
anonymous
  • anonymous
ohhh, i can pick any two numbers?
e.mccormick
  • e.mccormick
Well, 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
  • e.mccormick
I do see one big thing to be careful of in all of this.
e.mccormick
  • e.mccormick
"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!
e.mccormick
  • e.mccormick
\(1m^2=10000cm^3\)
anonymous
  • anonymous
thanks for the heads up !
e.mccormick
  • e.mccormick
Oops. 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.
anonymous
  • anonymous
I know, i still understood what you were saying :P
anonymous
  • anonymous
\[\large C=7998.49m^2\]
e.mccormick
  • e.mccormick
\(\cfrac{$15.50}{10000cm^2}\times 5.23cm^2\) The \(cm^2\) cancels....
anonymous
  • anonymous
15.50 x 1000= 15,500
e.mccormick
  • e.mccormick
So 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...
anonymous
  • anonymous
back of the book says $0.80 each
e.mccormick
  • e.mccormick
Hmmm... I must have converted wrong somewhere.
e.mccormick
  • e.mccormick
I don't see where though... odd.... because that would be like it was a linear conversion and this is a square conversion.
e.mccormick
  • e.mccormick
Ah! Remembered. R needs to go back into the formula!
e.mccormick
  • e.mccormick
\(\cfrac{1800}{r}+2\pi r^2\implies \cfrac{1800}{5.23}+2\pi (5.23)^2=????\) That makes much more sense!
e.mccormick
  • e.mccormick
Yes, that got me something that will round to 80 cents.
anonymous
  • anonymous
thanks !!
e.mccormick
  • e.mccormick
Yah, 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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