Question

A firm is able to sell 84,000 items with the unit price of $30. If the firm increases the
price to $32, the sales will drop 5,000 items

Calculate the maximum unit price that the firm can set in order to have positive
revenue

Answer 1

conclusion : for every increase of $2 in price, the sales drop by 5000

let the number of $2 increases be x
then price = 30 + 2x
sales = 84000 - 5000x

revenue = sales * price
= (84000 - 5000x)(30+2x)
= 2000(84 - 5x)(15 + x) , I factored 1000 from the first part, and 2 from 2nd
= 2000(1260 + 9x - 5x^2)
d(revenue)/dx = 2000(9 - 10x) = 0 for a max of revenue
x = 9/10

there should be (9/10) increases of $2 or the price should be $31.80

check:
at x = 1, cost = 32 , sales = 79000
rev = 32*79000 = 2,528,000
at x = .9, cost = 31.8, sales = 79500
<b>rev = 2,528,100</b>
at x = .5, cost = 31 , sales = 81500
rev = 2,526,500