Help on optimization problem

[ShootingStar16]

Ok so I'm reviewing for my math midterm right now and we had a sub the last math class I had and she didn't teach us anything, so we were on our own for these problems.

 

There is this one problem I'm really stuck on. 

 

The question:

 

"A cylindrical boiler with an open top is to be built from stainless steel with a copper bottom. […] Determine the most economical dimensions for the boiler if the volume is 5Ï€ m³. "

 

The answer should be: r=1 and h=5. I know that you look for the minimum for this but I keep getting this question wrong. 

 

Any help is much appreciated!

 

Thanks  

[kw0573]

Hi 
the [....] part is really important
never mind let me work on it

Edited January 17, 2016 by kw0573

[ShootingStar16]

Hi 

the [....] part is really important

never mind let me work on it

The […] part is "The price of copper is five times the cost of stainless steel." 

[kw0573]

sorry i have to go but yeah that is important. you need an open aluminum part and a disk of copper, msg me if you need any further assistance i'll get back to you tomorrow.

the aluminum part is just the curved surface

[CkyBlue]

If you're really stuck on textbook problems, try quote Googling the answer. More than occasionally you will find someone who has asked the same question. After confirming my answer with the a Yahoo answers website, I've come to this conclusion.

 

kw0573 made a crucial point in saying the [...] was important.

 

Here is what you know.

 

V=5(pi)m³=(pi)r²h   (volume of a cylinder)

h=5/r^{2                         (boundary condition) This is the relationship you need to establish between height and radius.}

 

The surface area is 2(pi)rh+(pi)r².

 

In order to account for the materials' costs, you reference that EVERYTHING is made of stainless steel. The equation for your surface area in terms of cost becomes

 

2(pi)rh+5(pi)r²=SA

2(pi)r(5/r²)+5(pi)r²=SA (plug in boundary condition)

10(pi/r)+5(pi)r²

 

Take the derivative WRT radius, and set it to 0 in order to find a minimum. 

 

d(SA)/dr= (-10(pi)/r²)+10(pi)r=0

d(SA)/dr=(-10(pi)/r²)=-10(pi)r

r=1

h=5/r² 

h=5

 

Hope that helps, don't know why the font changed in between O.o