Question

In an auditorium, a charity show is conducted in order to raise at least $3,750. The auditorium can accommodate up to 180 spectators. Tickets cost $15 for students and $25 for adults. Identify the system of inequalities and the corresponding graph that determine whether the charity will reach its goal.

Answer 1

See explanation

Explanation:

Let
`x,\ x\ge 0` be the number of students and
`y,\ y\ge 0` be the number of adults on the show.

1. Tickets cost $15 for students, so x student tickets cost $15x.

Tickets cost $25 for adults, so y adult tickets cost $25y.

Total cost of all tickets is $(15x + 25y).

The charity show is conducted in order to raise at least $3,750, thus

`15x+25y\ge 3,750`

2. The auditorium can accommodate up to 180 spectators, hence

`x+y\le 180`

3. We get the system of inequalities:

`\left\{\begin{array}{l}x\ge 0\\y\ge 0\\x+y\le 180\\15x+25y\ge 3,750\end{array}\right.`

Plot all solutions sets to each inequality and the common region is the solution set to the system of inequalities. This region is not empty, so the charity will reach its goal. For example, if they sell 50 students tickets and 125 adult tickets, they will raise
`\$(15\cdot 50+25\cdot 125)=\$3,875`