Put into equations

Question

the opera house sells an average of 1,700 tickets per night when the price of each ticket is $25. An employee of the opera house noticed that with every one dollar decrease in the cost of each ticket, the number of tickets sold increased by 200 tickets. Write and equation that could be used to determine the number of one dollar decrease, x, that would revenue of 55,000.

Answer 1

Demand (or quantity sold) is a function of (that is, it depends on) the price.

($25,1700) <--- 1700 tix sold when price is $25

the demand increases by 200 tix when the price decreases by $1, so the slope is -200/1 = -200

B = y - mx = 1700 - (-200)(25) = 1700 - (-5000) = 1700 + 5000 = 6700

y = -200 x + 6700 <--- demand y, price x

The revenue = demand * x = x(-200x + 6700) = R(x)

55000 = x( - 200x + 6700)

0 = -200x^2 + 6700x - 55000

0 = -2x^2 + 67x - 550

0 = 2x^2 - 67x + 550

x = 19.11 or x = 14.39 per quadratic formula