.
A manufacturer makes 2-minute, 6-minute, and 9-minute film developers.
Each ton of 2-minute developer requires 6 minutes in plant A and 24 minutes in plant B.
Each ton of 6-minute developer requires 12 minutes in plant A and 12 minutes in plant B.
Each ton of 9-minute developer requires 12 minutes in plant A and 12 minutes in plant B.
If plant A is available 10 hours per day and plant B is available 16 per day,
how many tons of each type of developer can be produced so that the plants are fully used
~~~~~~~~~~~~~~~~~~~
Let x = tons of the 2-minute developer produced,
y = tons of the 6-minute developer produced,
z = tons of the 9-minute developer produced.
Then time needed at plant A is 6x + 12y + 12z minutes.
Time needed at plant B is 24x + 12y + 12z minutes.
Equations for fully used regime
6x + 12y + 12z = 600 minutes, (1)
24x + 12y + 12z = 960 minutes. (2)
Subtract equation (1) from equation (2). You will get
18x = 960-600 = 360
x = 360/18 = 20 tons.
Next, substitute x = 20 back into equation (1). You will get
12y + 12z = 600 - 6*20 = 600 - 120 = 480.
The same equation will be if you substitute x = 60 into equation (2).
So, we, actually, have one single equation for unknown y and z
12y + 12z = 480. (3)
It has infinitely many solutions in real positive numbers.
For example, you may give y any value from 0 to 40; then
z = = 40 - y.
So, the problem has infinitely many solutions; based on the condition, it is IMPOSSIBLE
to extract a unique solution.
It is not surprising, since the basic system of equations (1), (2) has only two equations for 3 unknowns.
In order for the problem admits a unique solution, one more restriction should be added to the problem's formulation.
Solved.