Chapter 7.6, Problem 45E

To determine

To solve; the optimal amount which must be spent on every type of ad and optimal whole audience.

Answer to Problem 45E

Solution: There are $50$ pages allocated ( $\$1M/\$20000$ per page) and hence the best possible audience is:

  $N=50(5$ million readers)

  $N=250$ million

Explanation of Solution

Let

  $x=$ minutes of TV ads

  $y=$ pages of newspaper ads

Constraints:

  $\begin{array}{l}100000x+20000y\le 1000000\Rightarrow 5x+y\le 50\\ 100000x\le 800000\Rightarrow x\le 8\\ 20000y\ge 200000\Rightarrow y\ge 10\end{array}$

Objective function:

R= $100000x+20000y$

Evaluating objective function to determine minimum cost:

  $\begin{array}{l}(0,10):R=100000(0)+20000(10)=200000\\ (8,0):R=100000(8)+20000(0)=800000\\ (0,50):R=100000(0)+20000(50)=1000000\end{array}$

Which means there must be $0$ minutes of Tv ads and $10$ pages of newspaper ads so that budget must go to newspaper ad

Hence

Tv ads: $\$0$

Newspaper ads: $\$1000000$