Systems of inequalities and verticies

[intervade]

Ok here's my problem...

"A theater is presenting a program on drinking and driving for students and their parents. The proceeds will be donated to a local alcohol information center. Admission is $2.00 for parents and $1.00 for students. However the situation has two constraints: The theater can hold no more than 150 people and every two parents must bring at least one student. How many parents and students should attend in order to raise the maximum amount of money?"

I've done similar problems where I have a objective function and some constraints but I think my problem is that I'm having trouble setting up the actual problem based on the description.. Here is what I have:

x = parents and y = students
z = 2x + y
x + y <= 150
2x + y >= 3

The last constraint I have is 2x + y >= 3 and I'm not sure if that's correct but I figured 2 parents and 1 kid is 3 so if every parent has 1 kid that's 3. Also, the problem doesn't say I NEED to graph that equation, I can find some points of this graph without actually graphing it but I'm not sure if that's the best way to solve this problem. Some of the points I came up with just using the equation are (0, 150) and (3/2,0).

I really have no direction on this problem, and could use some help!

Thanks

 

[Deleted member 4993]

Ok here's my problem...

"A theater is presenting a program on drinking and driving for students and their parents. The proceeds will be donated to a local alcohol information center. Admission is $2.00 for parents and $1.00 for students. However the situation has two constraints: The theater can hold no more than 150 people and every two parents must bring at least one student. How many parents and students should attend in order to raise the maximum amount of money?"

I've done similar problems where I have a objective function and some constraints but I think my problem is that I'm having trouble setting up the actual problem based on the description.. Here is what I have:

x = parents and y = students
z = 2x + y
x + y <= 150
2x + y >= 3 <<<< That one is incorrect.

Having said that I don't know what would be the correct answer.

No single parent allowed?

Can a single parent come by himself/herself?

If only paired parents allowed - then the constraint "every two parents must bring at least one student" should mean

y ? x/2

The last constraint I have is 2x + y >= 3 and I'm not sure if that's correct but I figured 2 parents and 1 kid is 3 so if every parent has 1 kid that's 3. Also, the problem doesn't say I NEED to graph that equation, I can find some points of this graph without actually graphing it but I'm not sure if that's the best way to solve this problem. Some of the points I came up with just using the equation are (0, 150) and (3/2,0).

I really have no direction on this problem, and could use some help!

Thanks

 

[DrMike]

If "every two parents must bring at least one student" that means \(\displaystyle y\geq x/2\)

 

[soroban]

Hello, intervade!

I think I got it . . . I think . . .

A theater is presenting a program on drinking and driving for students and their parents.
The proceeds will be donated to a local alcohol information center.
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.
(2) Every two parents must bring at least one student.

How many parents and students should attend in order to maximum the revenue?


Here is what I have:

\(\displaystyle x\) = parents, \(\displaystyle y\) = students

Revenue: .\(\displaystyle z \:=\: 2x + y\)

Capacity: .\(\displaystyle x + y \:\leq\: 150\)

\(\displaystyle \text{For each two }x\text{'s, there must be at least one }y.\)

\(\displaystyle \text{Here's my inequality: }\;y \:\geq \:\frac{x}{2}\)
. . The number of students is at least half the parents.

The graph will look like this:

        |
    150 *
        |:*
        |:::*
        |:::::*
        |:::::::*           *
        |:::::::::*     *
        |:::::::::::*
        |:::::::*     *
        |:::*           *
      --*-----------------*----
        |                150

\(\displaystyle \text{The vertices are: }\;(0,0),\;(0,150),\;(100,50)\)


Edit: . DrMike beat me to it . . . *sigh*