hi,

my name's Jennifer and I'm in high school. I've been having a hard time trying to solve this problem: A certain university wishes to determine room and board fees for the next academic year. For the current year, the fee is $3600 per student and 1800 students are living in the residence halls. Past data suggests that for every $300 increase in the semester fee, 200 fewer students will choose to live in the dorms. There are also costs to the university associated with the residence halls. The fixed costs total $2,000,000 per semester. the variable costs are currently $1000 per student but will fall $100 per student for each decrease of 100 students. Your task is to help the university determine the optimal fee. Assume linear relationships between the number of students and the fee and between the number of students and the cost per student.

If you could help me with this problem, it would be great.
Thanks!

Hi Jennifer,
I assume tht the optimal fee means the fee that resulty in the largest possible profit for the university, where the profit is the revenue minus the cost. This is a "max-min" problem that you can solve using calculus once you have an expression for the profit.

Let x be the dollar increase in the semester fee that is charged by the residence hall, so the fee will be $(3600 + x). For every $300 dollar increase the number of students who choose to live in the dorms drops by 200. Thus the number of students who choose to live in the dorms drops by

^{x}/_{300} 200 = ^{2}/_{3} x
Thus the number of students who choose t live in the dorms is
1800 - ^{2}/_{3} x
Each of these students pays $(3600 + x) and hence the revenue for the university is
$(3600 + x) (1800 - ^{2}/_{3} x)
The costs for the university are 2,000,000 plus $1000 minus $100 dollars for each student who decides not to live in the dorms. Write an expression for the costs and then the profit is the revenue minus the costs.

Penny