Final answer:
The budget should be allocated between radio and TV advertising to maximize reach while respecting the constraints that no medium can exceed 80% of the total budget and each medium must have at least one ad. The cost of one radio and one TV ad must be subtracted from the total budget before calculating the additional ads based on their respective reaches and costs. Maximization could involve an iterative approach or linear programming to determine the optimal number of ads.
Step-by-step explanation:
To determine how the budgeted amount of $20,000 should be allocated between radio and TV advertising, we must respect the constraints specified: no medium can exceed 80% of the total budget, and each medium must have at least one ad. Given that a radio commercial costs $300 and a TV ad costs $2000, the maximum number of ads for each can be calculated using the 80% budget limit.
The 80% budget limit for each medium is 0.8 × $20,000 = $16,000. Therefore, the maximum expenditure on radio would limit us to $16,000 / $300 ≈ 53 commercials and for TV, $16,000 / $2000 = 8 ads. However, the goal is to maximize reach within the budget, so we'll also calculate the reach per ad.
For radio, the first ad reaches 5,000 people and each subsequent ad reaches an additional 2,000 people. For TV, the first ad reaches 4,500 people with each additional ad reaching 3,000 more people.
Solving this as an optimization problem with constraints can follow several steps. One approach would be to start with the minimum requirement of one ad for each medium and then add additional ads in the medium which provides the greatest additional reach per dollar spent, respecting both the individual budget constraint of 80% and the overall budget of $20,000.
The solution would involve setting up an equation to determine the number of ads for each medium that maximizes reach while staying within the budget and constraints, and this would likely involve iterative calculation or a linear programming method.