the lagrange multiplier equation
posted by sunny
A cylindrical oilstorage tank is to be constructed for which the following costs apply:
cost per square meter
metal for sides $30.00 (cost per square meter)
combined costs of concrete base and metal bottom $37.50(cost per square meter)
top 7.50 (cost per square meter)
The tank is to be constructed with dimensions such that the cost is minimum for whetever capacity is selected.
a) one possible approch to slecting the capacity is to build the tank large enough for an additional cubic meter of capacity to cost $8.(note that this does not mean $8 per cubic meter average for the entire tank.) what is the optimal diameter and optimal height of the tank?
b) instead of the approach used in part(a), the tank is to be of such a size that the cost will be $9 per cubic meter average for the entire storage capacity of the tank. set up the lagrange multiplier equations and verify that they are satisfied by an optimal diameter of 20 m and optimal height of 15m.

MathMate
b.
The cost function is given by:
C(r,h)=30(2πrh)+(37.5+7.5)πr²
=15π(4rh+3r²)
The volume, is given by
V(r,h)=πr²h
The "Lagrangian" is therefore obtained by including the constraint for a particular volume Vo,
Vo = πr²h
C(x,y) = 15π(4rh+3r²) + λ(πr&^sup2;hVo)
The first order conditions can be obtained by partially differentiating with respect to each of the variables, r and h, and equate to zero:
15π(4h+6r)+λ(2πrh) = 0 ...(1)
15π(4r) + λ(πr²) = 0 ...(2)
I will leave it to you to verify the optimal diameter and height.
Hint: λ should equal 6 in this case.
Also, verify the global unit cost per volume to be $9/m³.
Numerous references on the subject are available, for example:
http://www.slimy.com/~steuard/teaching/tutorials/Lagrange.html
http://www.economics.utoronto.ca/osborne/MathTutorial/ILMF.HTM 
Anonymous
answer no molum

amit
1200
Respond to this Question
Similar Questions

mathoptimal diameterand height
A cylindrical oilstorage tank is to be constructed for which the following costs apply: cost per square meter metal for ides $30.00 combined costs of concrete base and metal bottom $37.50(cost per square meter) top 7.50 (cost per … 
calculus
A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $4 per square foot and the metal for the sides costs $4 per square foot. Find the dimensions that minimize cost … 
Calculus
A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $2 per square foot and the metal for the sides costs $7 per square foot. Find the dimensions that minimize cost … 
Calculus AB/AP
A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. If building the tank costs $8 per square meter for the base and … 
calculus
A tank with a rectangular base and rectangular sides is to be open at the top. It is to be constructed so that its width is 4 meters and its volume is 36 cubic meters. Building the tank costs $10 per square meter for the base and $5 … 
math
Suppose that you are responsible for predicting the cost of constructing a new office building. You find that the first story costs 500 dollars per square meter of floor space, and the second story costs 600 dollars per square meter. … 
lagrange multipliers
A cylindrical oilstorage tank is to be constructed for which the following costs apply: cost per square meter metal for ides $30.00 combined costs of concrete base and metal bottom $37.50(cost per square meter) top 7.50 (cost per … 
calculus
A box is contructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $5 per square foot and the metal for the sides costs $2 per square foot. Find the dimensions that minimize cost … 
Calc 1
A rectangular storage container with an open top is to have a volume of 10 m3. The length of the base is twice the width. Material for the base is thicker and costs $13 per square meter and the material for the sides costs $10 per … 
Calculus
A rectangular storage container with an open top is to have a volume of 10 . The length of its base is twice the width. Material for the base costs $12 per square meter. Material for the sides costs $5 per square meter. Find the cost …