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

Question

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

1. Find the demand equation for tickets to the game. 
2. Find the revenue equation
3. Find how many people need to buy tickets in order to maximize revenue
4. Find the maximum revenue

Answer 1

It's best to represent the data as two points: (10, 3000) and (5, 8000).

x = ticket price (independent variable)

y = number of people attending the game (dependent variable)

Using the slope formula, we calculate (3000 - 8000)/(10 - 5) = -1000 for slope.

Now our equation is y = -1000x + b. We need to find the y-intercept (or number of people attending the game if ticket price is $0). Substitute the point (10, 3000) for (x, y) and solve for b.

3000 = -1000(10) + b --> b = 13,000

1). Demand equation: y = -1000x + 13,000

2). Revenue is price of ticket multiplied by quantity of tickets sold --> revenue = xy

3). Revenue is maximized at two points...when price = $6, revenue = $42,000. When price = $7, revenue = $42,000. (if you view a table of the points, you will see that at 6 and 7, the product is greatest)

4). Maximum revenue = $42,000.