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A company produces two types of computer models, X and Y. The production of each item of model [tex]$X$[/tex] requires 2 hours of assembly time, 1 hour of inspection time, and 2 cubic feet of storage space, while each item of model [tex]$Y$[/tex] requires 1 hour of assembly time, 1.5 hours of inspection time, and 3 cubic feet of storage space. The company has a maximum of 100 hours of assembly time, 85 hours of inspection time, and 30 cubic feet of storage space. The company targets a profit of Birr 40 from each item of model [tex]$X$[/tex] and Birr 35 from each item of model [tex]$Y$[/tex].

Which one of the following shows the constraint inequalities and the objective function, [tex]$Z_{\text{max}} = 40x + 35y$[/tex]?

A. [tex]$\left\{\begin{array}{l} 2x + y \geq 100 \\ x + 1.5y \geq 85 \\ 2x + 3y \leq 30 \\ x \geq 0, y \geq 0 \end{array}\right.$[/tex]

B. [tex]$\left\{\begin{array}{l} 2x + y \geq 100 \\ x + 1.5y \geq 85 \\ 2x + 3y \geq 30 \\ x \geq 0, y \geq 0 \end{array}\right.$[/tex]

C. [tex]$\left\{\begin{array}{l} 2x + y = 100 \\ x + 1.5y = 85 \\ 2x + 3y = 30 \\ x \geq 0, y \geq 0 \end{array}\right.$[/tex]

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]



Answer :

GinnyAnswer
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]