Questions: Question A company manufactures two types of cell phones, a basic model and a pro model. The basic model generates a profit of 100 per phone and the pro model has a profit of 150 per phone. On the assembly line the basic phone requires 7 hours, while the pro model takes 11 hours. The basic phone requires 1 hour and the pro phone needs 3 hours for finishing, which includes loading software. Both phones require 1 hour for testing. On a particular production run the company has available 1,540 work hours on the assembly line, 360 work hours for finishing, and 200 work hours in the testing department: What is that maximum profit from manufacturing both phones? Round your answer to the nearest whole number. Do not include a dollar sign or comma in your answer.

Question A company manufactures two types of cell phones, a basic model and a pro model. The basic model generates a profit of 100 per phone and the pro model has a profit of 150 per phone. On the assembly line the basic phone requires 7 hours, while the pro model takes 11 hours. The basic phone requires 1 hour and the pro phone needs 3 hours for finishing, which includes loading software. Both phones require 1 hour for testing. On a particular production run the company has available 1,540 work hours on the assembly line, 360 work hours for finishing, and 200 work hours in the testing department: What is that maximum profit from manufacturing both phones? Round your answer to the nearest whole number. Do not include a dollar sign or comma in your answer.
Transcript text: Question A company manufactures two types of cell phones, a basic model and a pro model. The basic model generates a profit of $\$ 100$ per phone and the pro model has a profit of $\$ 150$ per phone. On the assembly line the basic phone requires 7 hours, while the pro model takes 11 hours. The basic phone requires 1 hour and the pro phone needs 3 hours for finishing, which includes loading software. Both phones require 1 hour for testing. On a particular production run the company has available 1,540 work hours on the assembly line, 360 work hours for finishing, and 200 work hours in the testing department: What is that maximum profit from manufacturing both phones? Round your answer to the nearest whole number. Do not include a dollar sign or comma in your answer.
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Solution

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Solution Steps

To solve this problem, we need to set up a linear programming model. The objective is to maximize the profit, subject to the constraints on the available work hours for assembly, finishing, and testing. We will use the scipy.optimize.linprog function in Python to find the optimal solution.

Step 1: Define the Variables

Let \( x_1 \) be the number of basic phones produced and \( x_2 \) be the number of pro phones produced.

Step 2: Objective Function

The objective is to maximize the profit: \[ \text{Maximize } P = 100x_1 + 150x_2 \]

Step 3: Constraints

The constraints based on the available work hours are:

  1. Assembly hours: \( 7x_1 + 11x_2 \leq 1540 \)
  2. Finishing hours: \( x_1 + 3x_2 \leq 360 \)
  3. Testing hours: \( x_1 + x_2 \leq 200 \)

Additionally, the number of phones produced cannot be negative: \[ x_1 \geq 0 \] \[ x_2 \geq 0 \]

Step 4: Solve the Linear Programming Problem

Using the linear programming method, we find the optimal values for \( x_1 \) and \( x_2 \) that maximize the profit \( P \).

Step 5: Calculate the Maximum Profit

The optimal solution yields a maximum profit of: \[ P = 21750 \]

Final Answer

The maximum profit from manufacturing both phones is: \[ \boxed{21750} \]

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