Chapter 7.5, Problem 25E

To determine

To find: and graph a system of inequalities describing the different numbers of tickets that can be sold.

Answer to Problem 25E

The graph of system of inequalities describing the different number of tickets that can be sold is given below.

Explanation of Solution

Given information:

There are $30 reserved seat tickets and $20 general admission tickets. There are 2000 reserved seats available, and fire regulations limit the number of paid ticket holders to 3000. The promoter must take in at least $75000 in ticket sales.

Calculation:

Let, x be the number of reserved seat tickets and y be the number of general admission tickets.

Set up the inequalities:

For at least $75000 ticket sales $30x+20y\ge 75000$

Reserved seats available $x\le 2000$

Paid ticket holders limit of 3000 $x+y\le 3000$

Both tickets must be equal or greater than 0 which is $x\ge 0,y\ge 0$

The system of inequality is:

  $\begin{array}{l}30x+20y\ge 75000\\ x\le 2000\\ x+y\le 3000\\ x\ge 0\\ y\ge 0\end{array}$

Here $x\ge 0,y\ge 0$ represents the region above x-axis and on the right of y-axis including the axis. Also $x\le 2000$ represents the area on the left of the line $x=2000$ including the line $x=2000$

For $30x+20y\ge 75000$ take $30x+20y=75000$ make a table

  $\begin{array}{cccc}x& 0& 100& 2500\\ y& 3750& 3600& 0\end{array}$

Plot the points and join them

  

The region for $30x+20y\ge 75000$ will be above the line $30x+20y=75000$ because the region with point (0,0) is not included in the graph

For $x+y\le 3000$ take $x+y=3000$ make a table

  $\begin{array}{cccc}x& 0& 1000& 3000\\ y& 3000& 2000& 0\end{array}$

Plot the points and join them

  

The region for $x+y\le 3000$ will be below the line $x+y=3000$ because the region with point (0,0) is included in the graph

On combining all the above information the final graph with the required region looks like