To raise the maximum amount of​ money, ___ adults and ____ students should attend.

Question

please help me solve this math problem!

 

A theater is presenting a program on drinking and driving for students and their parents or other responsible adults. The proceeds will be donated to a local alcohol information center. Admission is $ 14.00 for adults and $ 7.00 for students. However, this situation has two constraints: The theater can hold no more than 150 people and for every two adults, there must be at least one student. How many adults and students should attend to raise the maximum amount of money?

Answer 1

In order to solve this, I would set up a matrix.

 

Let x = # of Adult Tickets to be sold

Let y = # of Student Tickets to be sold

 

Additional information:

 

(1)  Constraint of a total of 150 seats in the venue

(2)  There's a requirement of 2 adult tickets for to accompany each student ticket

 

Create math statements from the above:

 

x + y = 150 (total of tickets that can be sold, to maximize revenue)

 

x = 2y   (because the adult ticket count has to be double of the student count)

Bring the variables to the left of the equal, so that it states:  x - 2y = 0

 

I can now create the following matrix:

 

1          1          =          150

1         -2          =             0

 

Get rid of the value in R2, C1:   R1-R2 = new R2; which gives us the updated matrix of:

 

1         1         =          150

0         3         =            50

 

Next, get the R1, C2 value to 0; by the following:  R2-3R1 = new R1; which gives us the updated matrix of:

 

-3          0          =          -300

 0          3          =            150

 

Now, do the following operations, to get to 1x and 1y:

 

-1/3 R1 = new R1

  1/3 R2 = new R2

 

This gives us the new matrix of:

 

1         0         =          100

0         1         =            50

 

So, that gives us x=100; y=50

 

Therefore, we must sell 100 adult tickets, along with 50 student tickets, to get the maximum ticket sales revenue, because all 150 tickets were sold.

 

The maximum revenue is found out by:

 

14 * 100 = 1,400     [Adult ticket sales]

  7 *   50 =    350     [Student ticket sales]

 

The combined ticket sales are:  $1,400 + $350 = $1,750 [the maximum revenue is $1,750]