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Answer on Microeconomics Question for rae

Question #6124
a manufacturer charges $24 for headphones and has been selling 1000 of them per week. He estimated that for every $1 price reduction, 100 more headphones can be sold per week. What price should he charge for the headphones to maximize his total profit? Show your work.
Expert's answer
Assume x is price reduction. Let construct function of profit:
24 - x -price per one headphone,

also 1000+100x -number of sold headphones

then profit function looks like:

f = (24 - x) (1000 + 100x )

f = -100 x^2 + 1400x + 24000

we should find the value of x when this function has a maximum. For that take the first derivative:

f'= -200 x + 1400 in the point f'=0 we have maximum, so

-200x + 1400 = 0 and x = 7,

Thus the price should be 24 - 7 = 17 dollars

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