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February 26th, 2017, 10:05 PM   #1
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Smile 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.
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February 26th, 2017, 10:09 PM   #2
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Originally Posted by Bobbyjoe View Post
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.
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February 26th, 2017, 11:59 PM   #3
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Quote:
Originally Posted by Bobbyjoe View Post
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 $r_{min}= \sqrt[3]{\dfrac{45}{36}}$

Last edited by skipjack; February 27th, 2017 at 12:30 AM.
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February 27th, 2017, 06:05 AM   #4
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Quote:
Originally Posted by romsek View Post
$V=\pi r^2 h$

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

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

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

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

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

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 $r_{min}= \sqrt[3]{\dfrac{45}{36}}$
$Cost = (2r)^2(4.50) + \dfrac{2 V}{r}(2.50)$

am I solving for r or v?

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

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.
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