Scheduling Because Joe Slim's brother was recently elected to the State Senate, Joe's financial advisement concern, Inside Information Inc., has been doing a booming trade, even though the financial counseling he offers is quite worthless. $($None of his seasoned clients pays the slightest attention to his advice.$)$ Slim charges different hourly rates to different categories of individuals: $$5,000$ per hour for private citizens, $$50,000$ per hour for corporate executives, and $$10,000$ per hour for presidents of universities. Because of his taste for leisure, he feels that he can spend no more than $40$ hours$/$week in consultation. On the other hand, Slim feels that it would be best for his intellect were he to devote at least $10$ hours of consultation each week to university presidents. However, Slim always feels somewhat uncomfortable dealing with academics, so he would prefer to spend no more than half his consultation time with university presidents. Furthermore, he likes to think of himself as representing the interests of the common citizen, so he wishes to offer at least $2$ more hours of his time each week to private citizens than to corporate executives and university presidents combined. Given all these restrictions, how many hours each week should he spend with each tvpe of client to maximize his income?