Question

A telecom company manufactures mobile phones and landline phones. They require 9 hours to make a mobile phone and 1 hour to make a landline phone. The company can work not more than 1000 hours per day. The packing department can pack at most 600 telephones per day. If \(\mathrm{x}\) and \(\mathrm{y}\) are the sets of mobile phones and landline phones respectively then the inequalities are (1) \(x+y \geq 600,9 x+y \leq 1000, x \geq 0, y \geq 0\) (2) \(x+y \leq 600,9 x+y \geq 1000, x \geq 0, y \geq 0\) (3) \(x+y \leq 600,9 x+y \leq 1000, x \leq 0, y \leq 0\) (4) \(9 \mathrm{x}+\mathrm{y} \leq 1000, \mathrm{x}+\mathrm{y} \leq 600, \mathrm{x} \geq 0, \mathrm{y} \geq 0\)

Short answer

Answer: The correct set of inequalities is \(9x+y \leq 1000, x+y \leq 600, x \geq 0, y \geq 0\).

Step-by-step solution

Constraint 1: Hours per day

The company can work not more than 1000 hours per day. Since it takes 9 hours to make a mobile phone and 1 hour to make a landline phone, we can represent this constraint as: 9x + y ≤ 1000, where x and y are the number of mobile and landline phones respectively.

Constraint 2: Phones packed per day

The packing department can pack at most 600 telephones per day. So the total number of telephones made must be less than or equal to 600. This constraint can be represented as: x + y ≤ 600.

Constraint 3: Mobile and landline phones are non-negative

Since the number of mobile and landline phones cannot be negative, we have the following constraints: x ≥ 0 and y ≥ 0. Now, combining all three constraints, we get the following set of inequalities: 9x + y ≤ 1000 x + y ≤ 600 x ≥ 0 y ≥ 0 Comparing this with the given options, we find this matches with option (4). So the correct inequalities are: (4) \(9x+y \leq 1000, x+y \leq 600, x \geq 0, y \geq 0\)

Key concepts

Constraint Optimization

Constraint optimization involves finding the best solution to a problem given certain limits or restrictions. In the telecom company's case, they want to maximize their production of mobile and landline phones within two major constraints: time and packing capacity. The company can use up to 1000 hours of work in a day, and the packing department can handle 600 units per day. Given these conditions, the company needs to optimize the number of each type of phone produced, so they stay within the time and packing limits without exceeding available resources or capabilities.

When approaching constraint optimization problems, it's essential to:

• Identify the constraints, which are the limiting factors in the problem.
• Create equations or inequalities to represent these constraints.
• Determine an objective, such as maximizing or minimizing a certain quantity, to optimize.

The goal is to work within these constraints to find the best possible solution that meets the company's objectives.

Inequality Systems

A system of inequalities is a set of two or more inequalities that have common solutions. In the exercise, we have a system for the number of phones produced represented by:

• \(9x + y \leq 1000\)
• \(x + y \leq 600\)
• \(x \geq 0\)
• \(y \geq 0\)

Each inequality must hold true at the same time. The inequalities represent constraints such as hours worked and packing capacity.

Solving such systems involves finding values for \(x\) and \(y\) that satisfy all inequalities together. Graphical methods or algebraic techniques can be used to analyze these situations and pinpoint the solution set that satisfies every constraint. In our scenario, identifying the solution helps in determining how many phones can be produced per day while respecting the company's limits.

Mathematical Modeling

Mathematical modeling in this context refers to representing real-world problems using mathematical expressions. It is about creating equations or inequalities that capture essential details of a scenario. In the case of our telecom company, mathematical modeling involves interpreting the problem constraints into mathematical language to analyze and solve it efficiently.

Key steps in creating a mathematical model include:

• Defining the variables: Here, \(x\) represents mobile phones, and \(y\) represents landline phones.
• Formulating the equations or inequalities: We generated constraints on the manufacturing hours and the packing potential.
• Solving and interpreting the model to find meaningful results for decision-making.

The advantage of mathematical modeling is its ability to simplify complex problems, enabling better understanding and solutions.

Graphical Method

The graphical method is a way of solving systems of equations or inequalities by plotting them on a graph. It allows visualization of the solution area or points that satisfy all constraints simultaneously. For our inequalities:

• \(9x + y \leq 1000\)
• \(x + y \leq 600\)

You would plot these lines on a coordinate plane. The area where all constraints overlap is known as the feasible region, which contains all possible solutions.

This approach is especially helpful when dealing with two variables, as it visually represents the relationships and limits, making it easier to identify the optimal solutions. By observing the graphical intersection and the bounds, you can determine feasible production quantities within given constraints.

Feasible Region

The feasible region in a system of inequalities is the set of all possible points that satisfy all constraints. On a graph, it’s usually a polygonal area formed where the inequalities overlap. In the company's scenario, the feasible region is where the constraints: \(9x + y \leq 1000\), \(x + y \leq 600\), \(x\geq 0\), and \(y\geq 0\) intersect.

The boundaries of this region are influenced by the constraints, and only solutions within this area are considered valid for the problem.

• This area represents all combinations of mobile and landline phones that the company can produce while adhering to time and packing restrictions.
• Any point within this region is a viable solution, but the company will typically seek to find specific points that optimize the desired outcome, such as maximizing production, minimizing costs, etc.

Identifying the feasible region also helps in assessing the flexibility or limitations in fulfilling additional resource constraints if introduced.