To solve this problem, we will set up a linear programming model with the objective of maximizing the reach (number of people reached by the ads) under given constraints.

Step 1: Define Variables

Let:

• $x$ = number of radio ads
• $y$ = number of television ads

Step 2: Define the Objective Function

Each radio ad reaches 3,000 people, and each television ad reaches 7,000 people. Therefore, the total reach (in terms of people reached) is: $\text{Maximize } Z = 3000x + 7000y$

Step 3: Define Constraints

1. Budget Constraint: Each radio ad costs $200, and each television ad costs $500. The total budget is $40,000. $200x + 500y \leq 40000$

2. Minimum Ads Constraint: At least 10 ads of each type must be used. $x \geq 10$ $y \geq 10$

3. Radio Ads Greater Than or Equal to Television Ads: The number of radio ads must be at least as great as the number of television ads. $x \geq y$

Step 4: Solve the Problem

Now, let's solve this system by substituting values within the constraints to maximize $Z$.

Constraint Analysis:

From the budget constraint: $200x + 500y \leq 40000$ Dividing through by 100: $2x + 5y \leq 400$

To satisfy all constraints:

1. $x \geq 10$
2. $y \geq 10$
3. $x \geq y$
4. $2x + 5y \leq 400$

Let's examine potential values for $x$ and $y$ that satisfy these constraints and maximize the reach.

1. If $x = 50$ and $y = 50$:

   • Budget check: $200(50) + 500(50) = 10000 + 25000 = 35000$ (satisfies budget)
   • Reach: $3000(50) + 7000(50) = 150000 + 350000 = 500000$ people

2. If $x = 50$ and $y = 40$:

   • Budget check: $200(50) + 500(40) = 10000 + 20000 = 40000$ (exact budget)
   • Reach: $3000(50) + 7000(40) = 150000 + 280000 = 430000$ people

The configuration that maximizes the reach within the budget and constraints is:

• 50 radio ads and 40 television ads, reaching 430,000 people.

Solution

The campaign manager should use:

• 50 radio ads
• 40 television ads

This will reach a total of 430,000 people.

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Would you like further details or have questions about the solution process?

Follow-up Questions:

1. How would the solution change if the budget increased to $50,000?
2. What if the minimum required number of radio ads increased to 20?
3. How would reducing the cost of television ads to $400 impact the results?
4. What if the candidate wanted to prioritize a specific demographic by favoring television ads more?
5. How would adding a social media ad option with different costs and reach affect the campaign strategy?

Tip:

In linear programming problems, always check for feasibility first by ensuring that all constraints can be satisfied before optimizing the objective function.