$1-1.$ This exercise starts with a two$-$variable linear program similar in structure to the one of Sec$-$

tions $1\mathrm{.}1$ and $1\mathrm{.}2,$ but with a quite different story behind it$.$

$($a$)$ You are in charge of an advertising campaign for a new product, with a budget of $$1$ million. You can advertise on TV or in magazines. One minute of TV time costs $$20,000$ and reaches $1\mathrm{.}8$ million potential customers; a magazine page costs $$10,000$ and reaches $1$ million. You must sign up for at least $10$ minutes of TV time. How should you spend your budget to maximize your audi$-$ ence? Formulate the problem in AMPL and solve it$.$ Check the solution by hand using at least one of the approaches described in Section $1\mathrm{.}1.$

$($b$)$ It takes creative talent to create effective advertising; in your organization, it takes three person$-$weeks to create a magazine page, and one person$-$week to create a TV minute. You have only $100$ person$-$weeks available. Add this constraint to the model and determine how you should now spend your budget.

$($c$)$ Radio advertising reaches a quarter million people per minute, costs $$2,000$ per minute, and requires only $1$ person$-$day of time. How does this medium affect your solutions?

$($d$)$ How does the solution change if you have to sign up for at least two magazine pages? A maxi$-$ mum of $120$ minutes of radio?