1. A stadium has found that if ticket prices are $10 then 3000 people come to an event. If the tickets are priced at $5, then 8000 people come to the game. Assume that this relationship is linear.

Question

a) Find the ticket sales equation for tickets to the game.

 

Use t for the number of tickets and x for the price of each ticket. 

Answer 1

t=-1000x + 13,000 is the linear equation relating price x to tickets sold

or to simplify it, using units of 1,000, just t= -x + 13

slope =-1

t = -x + 13

(10,3), (8,5) = (x,t) where t = the number of tickets when multiplied by 1,000,

x = price in dollars

they are points with slope = -1 = (3-5)/(10-8) = -2/2 = -1 = m

t=mx + b

t=-(1)x + b

plug in either point to solve for b

5 = -8 + b

b=13 b is in units of 1,000

t =-x + 13 (or if you prefer to do this with thousands, then t =-1000x + 13,000)

f(x) =-x +13

at a price of 0, people will take 13,000 tickets. You can't give more away for free

at a price of $13, no one will buy tickets

revenue maximizing price is the midpoint of the line or $6.50

Revenue = tx = -x^2 +13x

R'(x) = -2 x + 13 = 0

-2x +13 = 0

x= 13/2

x= 6.5 = $6.50

max R = -(6.5)^2 +13(6.5) = -42.25+ 84.5 = 42.25 = $42,250

maximum revenue = $42,250 at price = $6.50 per ticket