Question

You want to sell a certain number n of items in order to
maximize your profit. Market research tells you that if you set the price at
$1.50, you will be able to sell 5000 items, and for every 10 cents you lower
the price below $1.50 you will be able to sell another 1000 items. Suppose
that your fixed costs (“start-up costs”) total $2000, and the per item cost of
production (“marginal cost”) is $0.50. Find the price to set per item and the
number of items sold in order to maximize profit, and also determine the
maximum profit you can get.

Answer 1

this is what i got so far:
r(x)=nx---> revenue for selling n items at x dollars
c(x)=2000+0.5n ---> cost
n(x)=5000 + 1000(1.5- 0.10x)
these are my equations, i calculated it until the end, but i got wrong answer.
i got x=1.55 or x=33.95
but the right answer is x=1.25

Answer 2

let y be the price of product
if n=5000 +1000x then y=1.5-0.1x
and our profit (that we want to maximize) is:
(5000+1000x)(1.5-0.1x-0.5)-2000
=(5000+1000x)(1-0.1x)-2000
=5000-500x+1000x-100x2-2000
=-100x2+500x+3000
(first derivative)->-200x+500=0
[second derivative: -200 (and we confirm that the function has maximum point)]
->x=-500/-200=2.5
price to set: 1.25
number of items: 7500
our profit: 3625