Linear Programming Theater Word Problem: Setting up Constraints

Slecker

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I'm having trouble understanding the second constraint.


"A theater is presenting a program on drinking and driving for students and their parents[...] admission is $2.00 for parents and $1.00 for students. However, the situation has two constraints: 1) The theater can hold no more than 150 people and 2) every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?"


Let x = number of students


y = number of parents


Since the question prompt wants the maximum amount of revenue, then the objective function is z = x + 2y


The theater can hold no more than 150 people so the first constraint is simply: x + y <= 150


The second constraint is, "every two parents must bring at least one student," but I don't understand how to model this.


I've looked up a solution and it said y <= 2x is how to model this constraint, but I don't understand why. If there must be at least one student for every two parents then why wouldn't the inequality be 2y >= x?
 
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I'm having trouble understanding the second constraint.


"A theater is presenting a program on drinking and driving for students and their parents[...] admission is $2.00 for parents and $1.00 for students. However, the situation has two constraints: 1) The theater can hold no more than 150 people and 2) every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?"


Let x = number of students


y = number of parents


Since the question prompt wants the maximum amount of revenue, then the objective function is z = x + 2y


The theater can hold no more than 150 people so the first constraint is simply: $x + y <= 150$


The second constraint is, "every two parents must bring at least one student," but I don't understand how to model this.


I've looked up a solution and it said y <= 2x is how to model this constraint, but I don't understand why. If there must be at least one student for every two parents then why wouldn't the inequality be 2y >= x?
If you write the constraint as

(x) >= (y/2) .................. (number of students must be greater than or equal to half of the number of parents)

Would that make more sense to you?
 
I'm having trouble understanding the second constraint.


"A theater is presenting a program on drinking and driving for students and their parents[...] admission is $2.00 for parents and $1.00 for students. However, the situation has two constraints: 1) The theater can hold no more than 150 people and 2) every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?"


Let x = number of students


y = number of parents


Since the question prompt wants the maximum amount of revenue, then the objective function is z = x + 2y


The theater can hold no more than 150 people so the first constraint is simply: $x + y <= 150$


The second constraint is, "every two parents must bring at least one student," but I don't understand how to model this.


I've looked up a solution and it said y <= 2x is how to model this constraint, but I don't understand why. If there must be at least one student for every two parents then why wouldn't the inequality be 2y >= x?
Let's see why 2y >= x works or doesn't work. We know that every two parents must bring at least one student. Say there are 10 parents so they must bring at least 5 students. Now 2(20) >= 5 simply is not true.
On the other hand if y<= 2x, then we get 10 <= 2(5) is true.
Also, the statement every two parents must bring at least one student leaves the opening for every two parents to bring 100 students so in the end 2y would NOT be bigger than x, since x is very large.
 
If you write the constraint as

(x) >= (y/2) .................. (number of students must be greater than or equal to half of the number of parents)

Would that make more sense to you?

But why half the parents? Could you elaborate a little more, please?
 
p = parent, s = student

p1 and p2 bring s1 : $5
p3 and p4 bring s2 : $5
...
p99 and p100 bring s50 : $5

Total collected: $250 = max
 
p = parent, s = student

p1 and p2 bring s1 : $5
p3 and p4 bring s2 : $5
...
p99 and p100 bring s50 : $5

Total collected: $250 = max
You left u's and v's and are now onto p's ans s's? What happens when you finish the alphabet?
 
Let's see why 2y >= x works or doesn't work. We know that every two parents must bring at least one student. Say there are 10 parents so they must bring at least 5 students. Now 2(20) >= 5 simply is not true.
On the other hand if y<= 2x, then we get 10 <= 2(5) is true.
Also, the statement every two parents must bring at least one student leaves the opening for every two parents to bring 100 students so in the end 2y would NOT be bigger than x, since x is very large.

But isn't 2(20) >= 5 true as 40 is larger than 5?

When I think about the relationship in terms of an equation it makes sense to me, such as y = 2x because for every child there must be 2 parents. However, this means that for every two parents there's ONLY one child and misses the fact that "for every two parents there must be AT LEAST one child." Since one child is the minimum it's implied that every two parents, or pair of parents, can bring more than just one child. At the very least I know it has to be an inequality simply based on the keyword "at least," so either y<=2x or y>=2x. But it's not a very convincing explanation, and that's what I'm looking for. I would like a convincing explanation as to why y<= 2x is the correct inequality to use within the context of the given situation.
 
p = parent, s = student

p1 and p2 bring s1 : $5
p3 and p4 bring s2 : $5
...
p99 and p100 bring s50 : $5

Total collected: $250 = max

Thanks for replying! I already know the solution, I've looked it up. I'm trying to understand why modeling the second constraint, "for every two parents they must bring at least one students," as y <= 2x is correct.
 
I'm having trouble understanding the second constraint.

"A theater is presenting a program on drinking and driving for students and their parents[...] admission is $2.00 for parents and $1.00 for students. However, the situation has two constraints: 1) The theater can hold no more than 150 people and 2) every two parents must bring at least one student. How many parents and students should attend to raise the maximum amount of money?"

Let x = number of students

y = number of parents

Since the question prompt wants the maximum amount of revenue, then the objective function is z = x + 2y

The theater can hold no more than 150 people so the first constraint is simply: x + y <= 150

The second constraint is, "every two parents must bring at least one student," but I don't understand how to model this.

I've looked up a solution and it said y <= 2x is how to model this constraint, but I don't understand why. If there must be at least one student for every two parents then why wouldn't the inequality be 2y >= x?

When I think about the relationship in terms of an equation it makes sense to me, such as y = 2x because for every child there must be 2 parents. However, this means that for every two parents there's ONLY one child and misses the fact that "for every two parents there must be AT LEAST one child." Since one child is the minimum it's implied that every two parents, or pair of parents, can bring more than just one child. At the very least I know it has to be an inequality simply based on the keyword "at least," so either y<=2x or y>=2x. But it's not a very convincing explanation, and that's what I'm looking for. I would like a convincing explanation as to why y<= 2x is the correct inequality to use within the context of the given situation.

Since English can be convoluted, often the way to figure out what a statement in English means mathematically is to change it step by step, being very careful:
every two parents must bring at least one student

for every two parents there is at least one student

there is at least one student for every two parents

the number of students is at least 1/2 the number of parents

x >= y/2

2x >= y

y <= 2x

Is there any step there at which you are unconvinced?

Now, answering your questions:

Yes, a pair of parents can bring more than one child; so the number of children can be greater than y/2. This is the same as saying that y/2 can be less than x; so the inequality will say y <= ..., not y >= ... .
 
I'm trying to understand why modeling the second constraint, "for every two parents they must bring at least one students," as y <= 2x is correct.
Why "model" anything?
It's evident that to get max revenue, as many parents as possible
are required, since they pay more than students.
So groups of 2 parents and 1 student are required.
150 / 3 = 50, @ $5 each = $250.
 
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Since English can be convoluted, often the way to figure out what a statement in English means mathematically is to change it step by step, being very careful:
every two parents must bring at least one student

for every two parents there is at least one student

there is at least one student for every two parents

the number of students is at least 1/2 the number of parents

x >= y/2

2x >= y

y <= 2x

Is there any step there at which you are unconvinced?

Now, answering your questions:

Yes, a pair of parents can bring more than one child; so the number of children can be greater than y/2. This is the same as saying that y/2 can be less than x; so the inequality will say y <= ..., not y >= ... .


Yes, thank you so much, Dr. Peterson! :grin:

That explanation was wonderfully simple and concise, and the key component for me was the statement, "the number of students is at least 1/2 the number of parents," followed by the inequality x >= y/2. It made everything click for me, and I see that taking x >= y/2 and simply solving/ isolating y gives the inequality y <= 2x.


 
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