$\backslash$begin$\{$tabular$\}\{|$l$|$c$|$c$|$c$|\}$

$\backslash$hline Constraint & Television & Radio & Online $\backslash \backslash$

$\backslash$hline Audience per advertisement & $400,000$ & $72,000$ & $160,000\backslash \backslash$

$\backslash$hline Cost per advertisement & $\backslash (\backslash$$ $2,000\backslash )$ & $\backslash (\backslash$$ $300\backslash )$ & $\backslash (\backslash$$ $600\backslash )\backslash \backslash$

$\backslash$hline Maximum media usage & $10$ & $20$ & $10\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$($a$)$ If the promotional budget is limited to $\backslash (\backslash$$ $20,500\backslash ),$ how many commercial messages should be run on each medium to maximize total audience contact?

$\backslash$begin$\{$tabular$\}\{$lll$\}$

Television & $\backslash \backslash$

Radio & $15\backslash \backslash$

Online & $10$

$\backslash$end$\{$tabular$\}$

What is the allocation of the budget among the three media, and what is the total audience reached?

$\backslash$begin$\{$tabular$\}\{$ll$\}$

Television Budget $($in dollars$)$ & $\backslash (\backslash$$ $\backslash )\backslash \backslash$

Radio Budget $($in dollars$)$ & $\backslash (\backslash$$ $500\backslash )\backslash \backslash$

Online Budget $($in dollars$)$ & $\backslash (\backslash$$ $8000\backslash )\backslash \backslash$

Total Audience & $5830000\backslash \backslash$

$\backslash$hline

$\backslash$end$\{$tabular$\}$

$($b$)$ By how much would audience contact increase if an extra $\backslash (\backslash$$ $100\backslash )$ were allocated to the promotional budget? $($Round your answer to the nearest whole number.$)$