Posted by Leo on Monday, August 4, 2014 at 11:39pm.
Maria, who has been a computer consultant for the past 5 years, has 100 regular customers. Her standerd fee is $90 per hour. She decided to raise her fees, and folowing some research and based on her previous experience, she expects that she will lose a customer for every $1 per hour increase in cost. What should be the new hourly fee for Maria to maximize her income?

math  Steve, Tuesday, August 5, 2014 at 12:00am
income = rate * customers
I see nothing about how many hours she works per customer, so I'll just assume she works one hour each. Actually, we will see that if she works the same constant number of hours for each customer, the number does not matter. If the billable hours were also affected by the rate, then we'd have a different function.
If she charges $x over $90, her income is thus
f(x) = (90+x)(100x) = 9000+10xx^2
This is just a parabola with the vertex (maximum income) at x=5.
So, max income occurs when she charges $95/hr.
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