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You work for a large tech company and have been put in charge of a new division tasked with developing cellphones to potentially compete with the iPhone. There are two models of phones your division will produce: the standard model and the deluxe model. Your supervisor tells you that if this new division does not prove profitable during the first quarter, the owner of the tech company may scrap the whole cell phone plan and your division with it. Therefore, it is up to you to generate the most possible revenue from the division.

The company informs you that one standard model will sell for [tex]\$62[/tex], and one deluxe model will sell for [tex]\$68[/tex]. If we let [tex]x[/tex] represent the number of standard models sold and [tex]y[/tex] represent the number of deluxe models sold, the revenue function is given by
[tex]\[
R(x, y) = 62x + 68y
\][/tex]

However, there are some constraints in the manufacturing process that you must be aware of:

Constraint 1: The owner has given you 8 workers and stated they must work no more than 460 total hours in a week. Since the standard model takes 3 hours to make, and the deluxe model takes 4 hours to make, you need to make sure that
[tex]\[
3x + 4y \leq 460
\][/tex]

Constraint 2: The boss is also a big believer in creating a strong work ethic, so the number of phones you produce must require a minimum of 360 hours a week. Therefore, you must ensure that
[tex]\[
3x + 4y \geq 360
\][/tex]

Constraint 3: The company has deals with electronic companies that allow acquiring parts for the deluxe phones cheaper than the standard phones. The parts for one standard phone cost [tex]\$20[/tex], and the parts for one deluxe phone cost [tex]\$10[/tex]. Since you have a weekly budget of [tex]\$1900[/tex], you must also ensure that
[tex]\[
20x + 10y \leq 1900 \Rightarrow 2x + y \leq 190
\][/tex]

Putting all this information together, your task is to maximize
[tex]\[
R(x, y) = 62x + 68y
\][/tex]
subject to the constraints
[tex]\[
\begin{aligned}
x &\geq 0, \\
y &\geq 0, \\
3x + 4y &\leq 460, \\
3x + 4y &\geq 360, \\
2x + y &\leq 190.
\end{aligned}
\][/tex]



Answer :

GinnyAnswer
To solve the problem of maximizing the revenue function [tex]\(R(x, y) = 62x + 68y\)[/tex] subject to the given constraints, we need to break down each step carefully:

### Step-by-Step Solution:

1. Identify the Objective Function:
Your objective is to maximize the revenue [tex]\(R(x, y)\)[/tex], which is given by:
[tex]\[ R(x, y) = 62x + 68y \][/tex]
where [tex]\(x\)[/tex] is the number of standard models sold and [tex]\(y\)[/tex] is the number of deluxe models sold.

2. List the Constraints:
You have several constraints to consider:
- The total work hours for the workers:
[tex]\[ 3x + 4y \leq 460 \][/tex]
- The minimum work hours required:
[tex]\[ 3x + 4y \geq 360 \][/tex]
This can be rewritten as:
[tex]\[ -3x - 4y \leq -360 \][/tex]
- The budget constraint on parts:
[tex]\[ 2x + y \leq 190 \][/tex]
- Non-negativity of the number of phones produced:
[tex]\[ x \geq 0, \quad y \geq 0 \][/tex]

3. Set Up the Linear Programming Problem:
To use the Linear Programming method, we need the constraints and the objective function:

- Maximize: [tex]\(62x + 68y\)[/tex]
- Subject to:
[tex]\[ \begin{aligned} 3x + 4y &\leq 460 \\ -3x - 4y &\leq -360 \\ 2x + y &\leq 190 \\ x &\geq 0 \\ y &\geq 0 \end{aligned} \][/tex]

4. Solve the Linear Programming Problem:
Using the simplex method or another linear programming solver, we solve the system of inequalities to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that maximize the revenue function.

5. Extract the Solution:
The optimal solution to the problem is:
[tex]\[ x = 60, \quad y = 70 \][/tex]

6. Calculate the Maximum Revenue:
Substitute the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] back into the revenue function:
[tex]\[ R(60, 70) = 62(60) + 68(70) \][/tex]
Calculate the products and sum them up:
[tex]\[ R = 62 \times 60 + 68 \times 70 = 3720 + 4760 = 8480 \][/tex]

### Conclusion:
The maximum revenue that can be achieved under the given conditions is [tex]\(\$8480\)[/tex] by producing and selling [tex]\(60\)[/tex] standard models and [tex]\(70\)[/tex] deluxe models.

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