In February, 2002, each episode of "Becker" was typically seen in 8.3 million homes, while each episode of "The Simpsons" was seen in 7.5 million homes. Your marketing services firm has been hired to promote Bald No More, Inc.'s, hair replacement process by buying at least 30 commercial spots during episodes of "Becker" and "The Simpsons". The cable company running "Becker" has quoted a price of $1700 per spot, while the cable company showing "The Simpsons" has quoted a price of $1300 per spot. Bald No More's advertising budget for TV commercials is $75000, and it would like no more than 50% of the total number of spots to appear on "The Simpsons". How many spots should you purchase on each show to reach the most homes?

Question

In February, 2002, each episode of "Becker" was typically seen in 8.3 million homes, while each episode of "The Simpsons" was seen in 7.5 million homes. Your marketing services firm has been hired to promote Bald No More, Inc.'s, hair replacement process by buying at least 30 commercial spots during episodes of "Becker" and "The Simpsons". The cable company running "Becker" has quoted a price of $1700 per spot, while the cable company showing "The Simpsons" has quoted a price of $1300 per spot. Bald No More's advertising budget for TV commercials is $75000, and it would like no more than 50% of the total number of spots to appear on "The Simpsons". How many spots should you purchase on each show to reach the most homes?

You should purchase spots on "Becker" and spots on "The Simpsons".

Answer 1

To determine how many commercial spots to purchase on each show, we need to consider two factors: the advertising budget and the goal of reaching the most homes.

First, let's consider the advertising budget of $75,000. We want to maximize the number of homes reached within this budget.

Next, let's consider the distribution of spots between "Becker" and "The Simpsons". The problem states that Bald No More would like no more than 50% of the total number of spots to appear on "The Simpsons".

To find the optimal number of spots on each show, we can use a two-step approach. First, we'll find the maximum number of spots we can purchase within the budget, assuming all spots are on one show ("Becker"). Then, we'll allocate spots between the two shows based on the distribution requirement.

Step 1: Finding the maximum number of spots on one show (Becker) within the budget:
- Cost per spot for "Becker" is $1700.
- Budget is $75,000.
- Maximum number of spots on "Becker" = Budget / Cost per spot for "Becker" = $75,000 / $1700 ≈ 44.11 spots.

Since we can't purchase a fractional number of spots, we can only purchase a maximum of 44 spots on "Becker" within the budget.

Step 2: Allocating spots between the two shows based on the distribution requirement:
- Total number of spots (including both shows) should be at least 30.
- The maximum number of spots we can purchase on "Becker" is 44 (as determined in Step 1).
- Let x represent the number of spots on "Becker".
- The number of spots on "The Simpsons" should be no more than 50% of the total number of spots.

Using these conditions, we can set up the following equation to solve for x:

x + y = 30 (Equation 1)
x + y ≤ 44 (Equation 2)
y ≤ 0.5 * (x + y) (Equation 3)

Solving this system of equations, we find x ≈ 19.57 and y ≈ 10.43.

Therefore, we should purchase approximately 20 spots on "Becker" and approximately 10 spots on "The Simpsons" to reach the most homes while staying within the advertising budget and distribution requirement.