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 February 26th, 2017, 10:05 PM #1 Member   Joined: Feb 2017 From: henderson Posts: 36 Thanks: 0 Confusing word problem A metal barrel is to be manufactured out of two different types of metal, one type for the top and bottom and one for the curved side. The metal for the curved side piece costs \$2.50 per square meter and the metal for the top and bottom costs \$4.50 per square meter. The top and bottom circles must be cut out of a square piece of metal whose side length is the diameter of the circle and the rest of the square is wasted (so contributes to the cost). If the volume is to be 9 cubic meters, find the dimensions of the barrel that minimizes the cost. Identify the interval you are minimized over and show your solution is the minimum. Last edited by skipjack; February 27th, 2017 at 12:24 AM. February 26th, 2017, 10:09 PM   #2
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Quote:
 Originally Posted by Bobbyjoe A metal barrel is to be manufactured out of two different types of metal, one type for the top and bottom and one for the curved side. The metal for the curved side piece costs \$2.50 per square meter and the metal for the top and bottom costs \$4.50 per square meter. The top and bottom circles must be cut out of a square piece of metal whose side length is the diameter of the circle and the rest of the square is wasted (so contributes to the cost). If the volume is to be 9 cubic meters, find the dimensions of the barrel that minimizes the cost. Identify the interval you are minimized over and show your solution is the minimum.
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Last edited by skipjack; February 27th, 2017 at 12:29 AM. February 26th, 2017, 11:59 PM   #3
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Quote:
 Originally Posted by Bobbyjoe A metal barrel is to be manufactured out of two different types of metal, one type for the top and bottom and one for the curved side. The metal for the curved side piece costs \$2.50 per square meter and the metal for the top and bottom costs \$4.50 per square meter. The top and bottom circles must be cut out of a square piece of metal whose side length is the diameter of the circle and the rest of the square is wasted (so contributes to the cost). If the volume is to be 9 cubic meters, find the dimensions of the barrel that minimizes the cost. Identify the interval you are minimized over and show your solution is the minimum.
$V=\pi r^2 h$

$h = \dfrac{V}{\pi r^2}$

$area_{tb} = (2r)^2$

$area_{side} = 2\pi r h = 2\pi r \dfrac{V}{\pi r^2} = \dfrac{2 V}{r}$

$Cost = area_{tb}(4.50) + area_{side}(2.50)$

$Cost = (2r)^2(4.50) + \dfrac{2 V}{r}(2.50)$

Now, find the solution of

$\left .\dfrac{dCost}{dr}\right|_{r=r_{min}}= 0$

check that $r$ is the minimum by ensuring that

$\left . \dfrac{d^2 Cost}{dr^2}\right|_{r=r_{min}} > 0$

I get $r_{min}= \sqrt{\dfrac{45}{36}}$

Last edited by skipjack; February 27th, 2017 at 12:30 AM. February 27th, 2017, 06:05 AM   #4
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 Originally Posted by romsek $V=\pi r^2 h$ $h = \dfrac{V}{\pi r^2}$ $area_{tb} = (2r)^2$ $area_{side} = 2\pi r h = 2\pi r \dfrac{V}{\pi r^2} = \dfrac{2 V}{r}$ $Cost = area_{tb}(4.50) + area_{side}(2.50)$ $Cost = (2r)^2(4.50) + \dfrac{2 V}{r}(2.50)$ Now, find the solution of $\left .\dfrac{dCost}{dr}\right|_{r=r_{min}}= 0$ check that $r$ is the minimum by ensuring that $\left . \dfrac{d^2 Cost}{dr^2}\right|_{r=r_{min}} > 0$ I get $r_{min}= \sqrt{\dfrac{45}{36}}$
$Cost = (2r)^2(4.50) + \dfrac{2 V}{r}(2.50)$

am I solving for r or v?

also for the last part am I using a derivative test? February 27th, 2017, 06:34 AM   #5
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Quote:
 Originally Posted by Bobbyjoe $Cost = (2r)^2(4.50) + \dfrac{2 V}{r}(2.50)$ am I solving for r or v? also for the last part am I using a derivative test?
you are solving for $r$

$V=9$

I just like to wait as long as possible before plugging numbers in.

Yes, this whole method is basically the derivative test for extreme points and then the second derivative test to ensure that the point(s) found are minima. Tags confusing, problem, word Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post CreepyPastaMansion Probability and Statistics 1 June 23rd, 2014 06:49 AM YoungMoustache Elementary Math 1 September 20th, 2012 07:01 PM kensclark15 Algebra 9 September 17th, 2012 05:22 PM matt93 Algebra 3 November 3rd, 2011 09:53 AM cherryperry Elementary Math 1 May 29th, 2010 02:29 PM

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