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Break-even point for a college

startup cost = 350,000, tuition 18,000 per student per year, charge to college 12,000 per student for the first 100 students enrolled.  How many students does the college need to enroll the first year to break even? If the college can enroll 75 students the first year, how much profit will it make?  If the college raised tuition to 24,000 it will reduce enrollment to 35 should the college consider this?
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1 Answer

Let x be number of students enrolled (x<101); revenue=18000x while cost=12000x+350000 (per-student cost + startup cost).
 
So the Profit function P(x) (rev-costs) (lets assume all coefficients are in 1000s here for simplicity): P(x):=(18-12)x-350
 
To break even, simply solve for x so that P(x) = 0; to find expected profit for a given number of students, simply plug x into P(x)
 
Now, how might we deal with the second part...?  Hint: what will change in the profit function, and (assuming this is based on comparing with 75 enrolled students) how will it affect the profit?
 
Hope this helps :)