Here is a link from Washington with helpful information on the subject:
http://www.cs.washington.edu/prospective_students/undergrad/admissions/direct_admission
If your Grandparents are listed as dependents on your parent’s taxes plus all the sibs I think you do have a shot at getting some need based FA from Cornell especially if your parents don’t have savings, investments, lots of home equity, or other properties.businesses owned by them. If you are comparing what Cornell will cost to instate tuition and living at home it won’t be close though. Are you willing to work hard in the summer and a few hours a week in school? You can cover around $5000 doing that per year. You could take on $5000ish in loans per year. Maybe after that it would be close enough assuming your parents have essentially no assets beyond the high income.
You can try Yale. Supposedly they are looking for quality admits interested in CS. Their finaid is better than Cornell’s