Question 1194184: Jordan has $100 to buy some comic books as Holiday gifts for his friends. His friends likes the Star Wars books which cost $12 each, but they also like the Marvel books which cost $5 each. He also wants to buy no fewer than 12 books.
1. What is the objective function?
2. What are the constraints?
3. Use Desmos.com to draw the feasible region, the region that is bound by the system of constraints.
4. How many of each kind of book should he buy to maximize the number of books?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Problem 1
x = number of Star Wars books
y = number of Marvel books
z = x+y = total number of books
The objective function is z = x+y
The goal is to maximize z based on the constraints mentioned in problem 2
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Problem 2
x = number of Star Wars books
12x = cost of all the Star Wars books only
y = number of Marvel books
5y = cost of all the Marvel books only
12x+5y = total cost of both types of books
Because Jordan has $100, this means the total cost (12x+5y) cannot exceed 100.
So either 12x+5y = 100 or 12x+5y < 100
In other words, one constraint is 
Another constraint is  since he wants to "buy no fewer than 12 books"
In other words, the smallest number of books Jordan wants to buy is 12 books
Meaning that 12 is the lowest  can go.
Since x and y refer to the number of books, we also know that they must be nonnegative whole numbers. Meaning that  and 
Summary:
The four constraints are
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Problem 3
Type each constraint separately into Desmos
Refer to this link to see what I mean
https://www.desmos.com/calculator/fxioytydqp
Admittedly, it is a very ugly graph of overlapped regions.
If we graphed only the first two constraints, then we'd have this
https://www.desmos.com/calculator/snqlgefzzz
Keep in mind we focus on the upper right quadrant where x,y are nonnegative.
Notice the red and blue regions overlap to form a triangle with the following vertices:
(0,20)
(0,12)
(5.714, 6.286)
You can click at the point of intersection to have the coordinates show up. Those decimal values are approximate.
You may need to move the window around to get a better look at the graph, depending on the size of your screen.
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Problem 4
We'll use those (x,y) corner points mentioned at the end of problem 3. Plug each of them, one at a time, into the objective function set up in problem 1.
Vertex = (0,20)
z = x+y
z = 0+20
z = 20
Vertex = (0,12)
z = x+y
z = 0+12
z = 12
Vertex = (5.714, 6.286)
z = x+y
z = 5.714+6.286
z = 12
This vertex isn't realistic since x and y must be nonnegative integers. However, we can still get an idea about it.
Clearly z = 20 is the largest result, showing that Jordan can buy the most books if he buys only Marvel comics at $5 each.
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