Question 1108140: At the local ball park, the team charges $5 for each ticket and expects to make 1,400 in concessions. The team must pay its players $2,000 and pay all other workers $1,600. Each fan gets a free bat that cost the team $3 per bat. How many tickets must be sold to break even?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! profit = revenue minus cost.
revenue is 5 dollars a ticket plus 1400 dollars in concessions.
the cost for the players is 2000 dollars and the cost for the other workers is 1600.
each person who buys a tickets gets a free bat that costs 3 dollars each.
let x equal the number of people who buy a ticket.
revenue = 1400 + 5x.
that's the revenue from the concessions plus 5 dollars for each ticket sold.
cost = 3600 + 3x
that's 2000 for the players and 1600 for the workers and 3 dollars for each bat given away.
profit = revenue minus cost.
profit = 1400 + 5x - (3600 + 3x)
simplify to get:
profit = 1400 + 5x - 3600 - 3x.
combine like terms to get:
profit = -2200 + 2x.
to find the break even point, set profit equal to 0.
set p = 0 and the profit equation becomes:
0 = -2200 + 2x.
add 2200 to both sides of this equation to get:
2200 = 2x
divide both sides of this equation by 2 to get:
1100 = x
they will break even when the number of tickets sold is 1100.
to make a profit, they have to sell more than 1100.
to take a loss, they have to sell less than 1100.
profit equation is, once again:
p = -2200 + 2x
when x = 1100, the equation becomes:
p = -2200 + 2*1100 which becomes p = -2200 + 2200 which becomes p = 0.
they didn't take a loss, but they didn't make a profit either.
when x > 1100, like say 1200, the equation becomes:
p = -2200 + 2 * 1200 which becomes p = -2200 + 2400 which becomes p = 200.
now they've made a profit of 200 dollars if they sold 1200 tickets.
make x < 1100, like say 800, the equation becomes:
p = -2200 + 2 * 800 which becomes p = -2200 + 1600 which becomes p = -600.
negative profit is a positive loss, therefore they took a loss of 600 if they only sold 800 tickets.
let's look at the profit equation when they sold 1200 tickets and break it down into its revenue and cost components.
p = -2200 + 2 * x becomes p = -2200 + 2400 which becomes p = 200.
revenue is 1400 + 5*1200 = 1400 + 6000 = 7400.
cost is 3600 + 3*1200 = 3600 + 3600 = 7200.
profit is revenue minus cost = 7400 - 7200 = 200.
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