optimization trouble

[sonniebeth22]

the problem is...

An industrial container is in the shape of a cylinder w/ hemispherical ends. If the container must hold 1,000 liters of fluid, determine the radius, r, and the length, l, that minimizes the amount of material used in the construction of the tank

so, basically i know that:
1000=pir^2h

and i have to use multivariables...but i'm kinda confused as to where the 1,000 comes into effect, b/c when i differentiate, it just goes away and become irrelevent....so i think i'm not doing it right, in fact i know i'm not, b/c once i went through the whole process, it was a saddle point...and help would be great

 

[Gene]

Not too bad start but...
You forgot the hemispherical ends. And you changed variables. The question says the length is l and you used h. Taking those two things into account your volume equation becomes
V = (4/3)pi*r^3+pi*r^2*l
But you want to minimize the material for which you can use the surface area. The area equation is
A = 4pi*r^2 + 2pi*r*l
That is what you want to differentiate.
The next thing is the 1000. That is liters but h and l would be meters or centimeters so you have to know that one liter is 1000 cubic centimeters and you have to change the 1000 liters into cubic meters or cubic centimeters, whichever you decide to use for h and l.
Then you can solve the volume equation for l and substitute that into the area equation before you do the differentiation. Then dA/dr = 0 will give the minimum material.

 

[Guest]

Set up a pair of equations as per the following:
Vol = 4/3 pi r^3 + 2pi r l (first part is the hemispherical ends, other part is the middle).

Let the above equation equal 1000.
Rearrange the eqn so you have the variable l by itself.

second equation is a surface area eqn.
write this equation out , but replace the variable l with the format from above.

To make this this material a min. amount you will then do the dy/dx of it and solve for the varialbe r (let dy/dx =0, as it is the min--ie a turning point)
in the above dy/dx should be more correctly shown as the form of:
d(surface area)/ dr

Hope this helps

 

[Guest]

Sorry rush of blood
fix the vol by using the correct vol for the cylinder pi r^2 l