SOLUTION: A factory manufactures two types of gadgets, regular and premium. Each gadget requires the use of two operations, assembly and finishing, and there are at most 12 hours available f

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Question 1208741: A factory manufactures two types of gadgets, regular and premium. Each gadget requires the use of two operations, assembly and finishing, and there are at most 12 hours available for each operation. A regular gadget requires 1 hour of assembly and 3 hours of finishing, while a premium gadget needs 4 hours of assembly and 5 hour of finishing. Due to other restrictions, the company can make at most 100 gadgets a day. If a profit of $20 is realized for each regular gadget and $30 for a premium gadget, how many of each should be manufactured to maximize profit?
Found 2 solutions by math_tutor2020, ikleyn:
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

x = number of regular gadgets
y = number of premium gadgets

Let's make a table to find out how long each process takes.
RegularPremiumTotal
Assemblyx4yx+4y
Finishing3x5y3x+5y

Based on the results in the last column, we can set up these constraints

This is because assembly and finishing have at most 12 hours for each operation.
"At most 12 hours" is the same as saying "12 hours or fewer".

Another constraint is x+y <= 100 because the factory can make at most 100 objects per day.


Then,

The last two constraints are there to ensure x,y are never negative.
Keep in mind that x,y are also integers.

That completes the system of inequalities for the constraints.
If you were to graph each of them on the same xy axis, then you'd get a mess of regions that overlap. The region where all areas overlap is what you're looking for.
That's often a standard approach that many teachers and textbooks will follow.

But perhaps a more efficient and cleaner graph would be to look at the opposite of each inequality mentioned.
Eg: The opposite of is
Graph the opposite of each and look where none of the shaded regions overlap. Look at the blank white space. This space is where the original system would overlap.

I'm following a process that tutor Theo is using in this similar question
https://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.1187223.html

For this problem, the final shaded region is marked as an X shown below.

The final shaded region is in the upper right quadrant and below the line 3x+5y=12.
If you pick a point in region X, then that point satisfies all of the original inequalities.

The shaded region is a triangle with the vertices
A = (0, 0)
B = (4, 0)
C = (0, 2.4)
Vertex B is found by solving the system of equations In other words, plug in y = 0 to get x = 4

Vertex C is found by solving the system of equations Plug in x = 0 to get y = 12/5 = 2.4
We don't have to worry about the lines x+4y=12 and x+y=100


Regular gadgets give a profit of $20 while premium gives a profit of $30.
The profit function is P = 20x+30y
The goal is to find the max value of P based on the constraints.

It turns out P is maxed when we're at a vertex.
Plug in each (x,y) coordinate to find various values of P.
Vertex(x,y)P
A(0,0)20x+30y = 20*0+30*0 = 0
B(4,0)20x+30y = 20*4+30*0 = 80
C(0,2.4)20x+30y = 20*0+30*2.4 = 72

The max profit is P = 80. It happens when you make x = 4 regular gadgets and y = 0 premium gadgets. This corresponds to vertex B.
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.

As I look at this problem, the condition seems strange to me. 

This restriction "at most 100 gadgets per day", actually, does not work, 
since it is overlayed by other restrictions on available time.


Even if suppose, that this restriction is a fictive essence, then the solution 
for the problem can be easily get MENTALLY.


Indeed, there is only three competitive cases:

    (1) 4 regular, 0 premium gadget,    giving 4*20 = 80 dollars profit;

    (2) 0 regular, 2 premium gadgets,   giving 2*30 = 60 dollars profit;

    (3) 2 regular, 1 premium gadgets,   giving 2*20 + 30 = 70 dollars profit.


My interior feeling is that the numbers in the problem are INCORRECT.


Last several days, I see several posts at the forum with incorrect input data.

My impression is that in the opposite end of the internet there is a person,
which specially/intently corrupts data in incoming posts.

    This my message is to the  MANAGERS  of this project.
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