To solve the problem of modeling the restrictions on the woodworking artist's time and material costs as a system of inequalities, we need to carefully translate the given constraints into mathematical inequalities.
Let's break down the problem step-by-step:
### 1. Time Constraint
The artist spends:
- 3 hours for each type [tex]\(X\)[/tex] carving.
- 2 hours for each type [tex]\(Y\)[/tex] carving.
The total time he can spend in a week is limited to 36 hours. Therefore, the inequality representing this constraint is:
[tex]\[ 3x + 2y \leq 36 \][/tex]
### 2. Cost Constraint
The cost of materials is:
- \[tex]$4 for each type \(X\) carving.
- \$[/tex]5 for each type [tex]\(Y\)[/tex] carving.
The total cost for materials each week must not exceed \$100. Therefore, the inequality representing this constraint is:
[tex]\[ 4x + 5y \leq 100 \][/tex]
Putting these two constraints together, we have the system of inequalities:
[tex]\[ \begin{cases}
3x + 2y \leq 36 \\
4x + 5y \leq 100
\end{cases} \][/tex]
### Conclusion
Based on the provided options, the correct system of inequalities that models this situation is:
[tex]\[ \begin{cases}
3x + 2y \leq 36 \\
4x + 5y \leq 100
\end{cases} \][/tex]
Therefore, the correct answer is:
C. [tex]\( 3x + 2y \leq 36, 4x + 5y \leq 100 \)[/tex]
