This is a linear programming problem. Let’s define the variables, constraints, and objective function to solve it:

Variables:

• Let $x_1$ be the amount (in grams) of ingredient 1 used.
• Let $x_2$ be the amount (in grams) of ingredient 2 used.

Objective Function:

We aim to minimize cost, given as: $\text{Cost} = 6x_1 + 4x_2$

Constraints:

1. Vitamin A requirement: At least 12 units. $3x_1 + 0.5x_2 \geq 12$
2. Vitamin B requirement: At least 16 units. $2x_1 + x_2 \geq 16$
3. Non-negativity of variables: $x_1 \geq 0, \, x_2 \geq 0$

Steps:

1. Graph the feasible region defined by the constraints.
2. Identify corner points of the feasible region.
3. Evaluate the objective function at each corner point to find the minimum cost.

Let me calculate this. numpy as np from scipy.optimize import linprog

Coefficients of the objective function (minimize cost: 6x1 + 4x2)

c = [6, 4]

Coefficients for the inequality constraints

A = [[-3, -0.5], # -3x1 - 0.5x2 <= -12 (Vitamin A constraint) [-2, -1]] # -2x1 - x2 <= -16 (Vitamin B constraint)

Right-hand side values for the inequalities

b = [-12, -16]

Bounds for x1 and x2 (non-negative)

x0_bounds = (0, None) # x1 >= 0 x1_bounds = (0, None) # x2 >= 0

Solve the linear programming problem

result = linprog(c, A_ub=A, b_ub=b, bounds=[x0_bounds, x1_bounds], method='highs')

resultThe optimization was successful. The results indicate:

• Amount of ingredient 1 ($x_1$): 8 grams
• Amount of ingredient 2 ($x_2$): 0 grams
• Minimum cost: 48 pence

This means the minimum cost of producing the vitamin capsule is achieved by using only 8 grams of ingredient 1, with no need for ingredient 2.

Would you like a step-by-step breakdown or more details about the method used?

Related Questions:

1. How do we derive the constraints from the problem statement?
2. What is the geometric interpretation of the solution?
3. What happens if the cost of ingredient 2 changes?
4. How can we verify the solution is optimal?
5. What alternative methods can solve such linear programming problems?

Tip: Always check that your constraints form a feasible region; otherwise, the problem may not have a solution.