Chapter 7.1, Problem 62E

To determine

To calculate: the optimal amount that should be spent on each type of ad and the optimal total audience.

Answer to Problem 62E

The optimal audience will be 250 million for which the budget should be only for newspaper ads.

Explanation of Solution

Given information:

A company has maximum budget of $1000000 for national advertising of an allergy medication. Each TV ad costs $100000 and one page newspaper ad costs $20000. Each TV ad is to be viewed by 20 million viewers, and each newspaper ad is to be seen by 5 million readers. The company’s marketing department recommends that at most 80% of the budget be spent on TV ads.

Calculation:

Let, x be the minutes on TV ads and y be the pages on newspaper ads.

Set up the constraints or inequalities:

Total budget will be

  $\begin{array}{l}100000x+20000y\le 1000000\\ \Rightarrow 5x+y\le 50\end{array}$

For 80% on TV ads

  $\begin{array}{l}100000x\le 800000\\ x\le 8\end{array}$

For 20% on newspaper ads

  $\begin{array}{l}20000y\ge 200000\\ \Rightarrow y\ge 10\end{array}$

Objective function is $C=100000x+20000y$

Graph the inequalities

  

Evaluate the objective function at each vertex to determine optimal cost.

  $\begin{array}{l}\left(0,10\right):C=100000\left(0\right)+20000\left(10\right)=200000\left(\text{Minimum}\right)\\ \left(8,0\right):C=100000\left(8\right)+20000\left(0\right)=800000\\ \left(0,50\right):C=100000\left(0\right)+20000\left(50\right)=1000000\end{array}$

This means that there should be 0 minutes for TV ads and 10 pages for newspaper so all the budget should go to newspaper ad.

Therefore, TV ads: $0

Newspaper ads: $1000000

This also means that there will be 50 pages allocated $\left(\$1M/20000\right)$ so the optimal audience is $N=50\left(5\text{millionreaders}\right)=250$ million

Conclusion:

The optimal audience will be 250 million for which the budget should be only for newspaper ads.