To solve this problem, let's break down the constraints based on the maximum resources available:
1. Assembly Time Constraint:
- For model [tex]\(X\)[/tex], each unit requires 2 hours.
- For model [tex]\(Y\)[/tex], each unit requires 1 hour.
- The total maximum available assembly time is 100 hours.
This leads to the constraint:
[tex]\[
2x + y \leq 100
\][/tex]
2. Inspection Time Constraint:
- For model [tex]\(X\)[/tex], each unit requires 1 hour.
- For model [tex]\(Y\)[/tex], each unit requires 1.5 hours.
- The total maximum available inspection time is 85 hours.
This leads to the constraint:
[tex]\[
x + 1.5y \leq 85
\][/tex]
3. Storage Space Constraint:
- For model [tex]\(X\)[/tex], each unit requires 2 cubic feet.
- For model [tex]\(Y\)[/tex], each unit requires 3 cubic feet.
- The total maximum available storage space is 30 cubic feet.
This leads to the constraint:
[tex]\[
2x + 3y \leq 30
\][/tex]
4. Non-negativity Constraints:
- The number of units [tex]\(x\)[/tex] and [tex]\(y\)[/tex] cannot be negative since they represent quantities of computers produced.
This leads to the constraints:
[tex]\[
x \geq 0 \text{ and } y \geq 0
\][/tex]
Combining all these constraints, we get the system of inequalities:
[tex]\[
\left\{
\begin{array}{l}
2x + y \leq 100 \\
x + 1.5y \leq 85 \\
2x + 3y \leq 30 \\
x \geq 0 \\
y \geq 0
\end{array}
\right.
\][/tex]
This set of constraints corresponds to option D. So, the correct answer is:
D. [tex]\(\left\{
\begin{array}{l}
2x + y \leq 100 \\
x + 1.5y \leq 85 \\
2x + 3y \leq 30 \\
x \geq 0 \\
y \geq 0
\end{array}
\right.\)[/tex]
