University of Alabama in particular has massively increased the amount of merit aid it awards to out-of-state students, and it now attracts a huge number of students from the Northeast and the Chicago area that would never have considered it before. There are other schools using similar strategies.
As I mentioned in another comment, the cost of any given school is very unpredictable until you actually get accepted and get your financial package. The winning strategy is to research schools which are known for generous financial offers for similar students, and then apply to as many of them as is manageable to increase your odds of a winning "lottery ticket".
Now that I'm older I still get no money, but I don't even get the "merit scholarships."
My parents income was slightly above your income, yet I got Pell grants all four years.
No one at Rice has asked for my opinion, and I'm not privy to all the numbers behind the scenes, but I'm surprised they can't be more generous with the middle class since we have such a large endowment. They recently announced a big new financial aid initiative, but I wish tuition could be free again like it was until 1965!
https://edition.cnn.com/2018/09/18/health/rice-university-fi...
Early 90's, I had just left the military and was thus able to apply for financial aid without recording my parents' income. So I was able to claim having made $15k the year before. I also accurately reported my savings. My total non-loan financial aid package consisted of $500/year of some state matching fund. I was also expected to contribute 100% of my savings in year one. I was told I had made too much money for a Pell grant. Something which remained true the next year when my income totaled $8k.
So in 2.5 years of college after the military, I managed to deplete my 4 years of aggressive savings and take on $15k in debt, over the GI Bill.
All of this is anecdotal, but as I prepare for my son to go to college next year, I have no realistic hope of him getting a cent. $200k for four years at Rice seems reckless.