
Linear Programming
I understand the problems that give various constraints and ask to maximize or minimize something, but not this type.
A chem manufacturer wants to rent a fleet of 24 tank cars with a combined carrying capacity of 504,000 gallons. Tank cars with three capacities are available: 7000 gallons, 14,000 gallons, 28,000 gallons. How many of each should be leased?
Set x, y and z as the three different capacity cars.
x+y+z=24
7000x+14000y+28000z=504000
There are various answers. For example it can be 18 z's, or 17 z's and 2 y's, or 17z's and 4 x's.
The answer asks to express it as:
x=_t+_ y=_t+_ and z=t where _<=t<=_
that's something times t plus something, etc. where something <=t<=something

Re: Linear Programming
Determined the answer.
7000 gallon car = x1
14000 gallon car = x2
28000 gallon car = x3
Two equations from the given information:
x1+x2+x3=24 The number of the combined tank cars must be 24
7000x1+14000x2+28000x3 = 504000 Their combined capacity must be 504000
There are two equations with three unknowns. That can't be solved unless we select a value for one variable.
Call x3=t. Now solve for the other two variables in terms of t.
x1+x2+t=24
x1=24x2t
7000x1+14000x2+28000t=504000
7000(24x2t)+14000x2+28000t=504000
1680007000x27000t+14000x2+28000t=504000
7000x2+21000t=336000
7000x2=33600021000t
x2=483t= 3t + 48
x1+483t+t=24
x12t=24
x1=2t24
To find the constraints on t, set each equation (x1 and x2 in terms of t) >= 0. Then solve for t.
x1 = 2t24 >= 0
2t>=24
t>=12
x2=3t+48>=0
3t>=48
t<=16
So t is in the range 12<=t<=16
t>=0, but that's redundant.
So the answer is B.
x1 = 2t24
x2 = 3t+48
x3=t
12<=t<=16

Re: Linear Programming
Only 3 solutions if at least one of each:
2,9,13
4,6,14
6,3,15