In the following exercises, estimate the volume of the solid under the surface z = f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 4. f ( x , y ) = cos x + cos y , R = [ 0 , π ] × [ 0 , π 2 ]
In the following exercises, estimate the volume of the solid under the surface z = f(x. y) and above the rectangular legion R by using a Riemann sum with m = n = 2 and the sample points to be the lower left corners of the subrectangles of the partition. 4. f ( x , y ) = cos x + cos y , R = [ 0 , π ] × [ 0 , π 2 ]
can you solve this question and explain the steps used along the way
You manage a chemical company with 2 warehouses. The following quantities of
Important Chemical A have arrived from an international supplier at 3 different
ports:
Chemical Available (L)
Port 1
400
Port 2
110
Port 3
100
The following amounts of Important Chemical A are required at your warehouses:
Warehouse 1
Warehouse 2
Chemical Required (L)
380
230
The cost in£to ship 1L of chemical from each port to each warehouse is as follows:
Warehouse 1 Warehouse 2
Port 1
£10
Port 2
£20
Port 3
£13
£45
£28
£11
(a) You want to know how to send these shipments as cheaply as possible. For-
mulate this as a linear program (you do not need to formulate it in standard
inequality form) indicating what each variable represents.
(b) Suppose now that all is as in the previous question but that only 320L of
Important Chemical A are now required at Warehouse 1. Any excess chemical
can be transported to either Warehouse 1 or 2 for storage, in which case the
company must pay only the relevant transportation…
Suppose we have a linear program in standard equation form
maximize cx
subject to Ax = b,
x > 0.
and suppose u, v, and w are all optimal solutions to this linear program.
(a) Prove that z = u+v+w is an optimal solution.
(b) If you try to adapt your proof from part (a) to prove that that u+v+w
is an optimal solution, say exactly which part(s) of the proof go wrong.
(c) If you try to adapt your proof from part (a) to prove that u+v-w is an
optimal solution, say exactly which part(s) of the proof go wrong.
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