A concert is soon holding and the venue can hold 12000 people. The minimum price is 100 and all tickets can be sold out if the price of each ticket is set to this minimum. For every increment of 1 in the price, the number of tickets sold decreases by 20. It is given that the total expenditure of the performance is 1000000. Let x be the price of each ticket and $y be the overall profit of the concert.
(a) Show that the number of tickets sold at $x is 14000-20x.
(b) Show that y=-20x^2 +14000x -1000000.
(c) Write down the domains of the function in (a).
(d) Find the range of prices of each ticket such that there is an overall profit.
(e) Find the price of each ticket such that the profit is at maximum and the number of ticket sold at this price.
Assuming that the venue can hold 14,000 people ( not, 12,000 people)
(a) Let x be the number of incremental increases
So the number of tickets sold = (14000 - 20x)
We have that
Profit = Revenue - Cost of staging concert
Revenue = Number of people * Price
Let x be the number of incremental price increases......so....
Revenue = (14000 - 20x) ( 100 + x) simplify
Revenue = 1400000 - 2000x + 14000x - 20x^2 = -20x^2 + 12000x + 1400000
So.....the total profit is given by
Revenue - Expenses...so.....
y = -20x^2 + 12000x + 1400000 - 1000000
y= -20x^2 + 12000x + 400000
(c) See the graph, here : https://www.desmos.com/calculator/kqdbrpbuuz
The domain is [0, ≈ 632 ]
(d) The range of prices that will guarantee a profit =
(100 + 0 , ≈ 100 + 632) = [100, ≈732 ] (in dollars)
(e) Looking at the graph again.......the max occurs at (300, 2,200,000)
So the price that will guarantee a max = (100 + 300) = $400
And the number of tickets sold at this price = 14000 - 20(300) = 8000 tickets