This is a linear programming optimization problem where we aim to maximize the number of people reached given a set of constraints.

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Definitions

Let:

• $x$: Number of radio ads.
• $y$: Number of television ads.

Objective Function

The goal is to maximize the number of people reached: $Z = 3000x + 7000y$

Constraints

1. Budget constraint: $200x + 500y \leq 40,000$
2. At least 10 ads of each type: $x \geq 10, \quad y \geq 10$
3. The number of radio ads must be at least as great as the number of television ads: $x \geq y$

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Solving the Problem

I will calculate the optimal values of $x$ and $y$ that maximize $Z$ while satisfying all the constraints.from scipy.optimize import linprog

Coefficients for the objective function (maximize Z = 3000x + 7000y)

c = [-3000, -7000] # Negative for maximization in linprog (minimizes by default)

Coefficients for the constraints

A = [ [200, 500], # Budget constraint [-1, 0], # x >= 10 -> -x <= -10 [0, -1], # y >= 10 -> -y <= -10 [-1, 1] # x >= y -> -x + y <= 0 ] b = [40000, -10, -10, 0] # RHS of constraints

Bounds for x and y

x_bounds = (10, None) # x >= 10 y_bounds = (10, None) # y >= 10

Solve the linear programming problem

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

Extract results

result_x = result.x # Values of x and y objective_value = -result.fun # Maximum number of people reached

result_x, objective_value(array([175., 10.]), 595000.0)(array([175., 10.]), 595000.0)successThe optimal solution is:

• Number of radio ads ($x$): 175
• Number of television ads ($y$): 10
• Maximum number of people reached: 595,000

This solution satisfies all the constraints and uses the budget effectively.

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Explanation:

1. The budget of $40,000 is fully utilized: $200(175) + 500(10) = 40,000$
2. The conditions $x \geq 10$, $y \geq 10$, and $x \geq y$ are all met.

Would you like more details or clarification on any part of this solution?

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Related Questions:

1. What if the budget was increased to $50,000? How would the solution change?
2. What happens if the minimum number of ads for each type is raised to 15?
3. How does the solution change if television ads reach 10,000 people instead of 7,000?
4. Can you analyze the sensitivity of the solution to changes in the cost of ads?
5. How would the result differ if the goal was to minimize the cost for a fixed audience size?

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Tip:

In linear programming problems, always check whether the constraints are tight (fully used) in the optimal solution, as this can guide adjustments in future scenarios.