Questions: A company produces two products, A and B. - Product A requires 3 kg of metal and 6 minutes of labor. - Product B requires 4 kg of metal and 3 minutes of labor. - Product A makes a profit of 2.00. - Product B makes a profit of 1.50 - 1000 kg of metal is available. - 20 hours of labor are available. How many of product A and B should be made to maximize profits?

Solution Steps

To solve this problem, we need to set up a linear programming model. The objective is to maximize the profit, which is a function of the number of products A and B produced. The constraints are based on the availability of metal and labor. We will use the scipy.optimize.linprog function to find the optimal solution.

Let \( x \) be the number of product \( A \) produced and \( y \) be the number of product \( B \) produced.

The profit from producing products \( A \) and \( B \) can be expressed as: \[ P = 2x + 1.5y \] We aim to maximize \( P \).

The constraints based on the availability of metal and labor are:

1. Metal constraint: \( 3x + 4y \leq 1000 \)
2. Labor constraint: \( 6x + 3y \leq 1200 \)

The optimal solution found is: \[ x \approx 120, \quad y \approx 160 \] This means approximately 120 units of product \( A \) and 160 units of product \( B \) should be produced.

The maximum profit can be calculated as: \[ P = 2(120) + 1.5(160) = 240 + 240 = 480 \]

Final Answer