Question

48 passengers. An adult ticket costs $16 and a child ticket costs $8. The bus company must collect at least $560 to make a profit. If a represents the number of adult tickets sold and c represents the number of child tickets sold, which system of inequalities models this situation if they make a profit?

Answer 1

To model the situation with a system of inequalities, we start by establishing the variables and the constraints given in the problem:

- Let \( a \) represent the number of adult tickets sold.
- Let \( c \) represent the number of child tickets sold.

### 1. Total Passengers Constraint
The total number of passengers is given as 48, which gives us the first inequality:

\[
a + c \leq 48
\]

### 2. Revenue Constraint
The bus company must collect at least $560 to make a profit. The revenue from adult tickets is \( 16a \) (since each adult ticket costs $16), and the revenue from child tickets is \( 8c \) (each child ticket costs $8). Therefore, the second inequality for the revenue constraint is:

\[
16a + 8c \geq 560
\]

### 3. Non-negativity Constraints
Since it's impossible to sell a negative number of tickets, both \( a \) and \( c \) must be non-negative integers. This gives us the following inequalities:

\[
a \geq 0
\]
\[
c \geq 0
\]

### Summary of the System of Inequalities
Putting it all together, the system of inequalities that models this situation is:

\[
\begin{align*}
1. & \quad a + c \leq 48 \\
2. & \quad 16a + 8c \geq 560 \\
3. & \quad a \geq 0 \\
4. & \quad c \geq 0 \\
\end{align*}
\]