The Summer Theatre charges $4 per ticket, and it sells an average of 400 tickets nightly. The manager estimates that the ticket sales would decrease by 50 for each $1 increase in the ticket cost. What is the most profitable price to charge?
profit = revenue-cost
If there are x price increases,
revenue=(4+x)(400-50x)
actually, you don't mention cost, but that doesn't matter, since it will not change with the price.
So, to get maximum revenue, just find the vertex of
y=(4+x)(400-50x)=1600+200x-50x^2
it will occur at x = -b/2a = 2
so, the price will be 4+2= $6
To determine the most profitable price to charge, we need to find the ticket price that maximizes revenue. Revenue is calculated by multiplying the number of tickets sold by the ticket price.
Let's start by calculating the revenue at the initial ticket price of $4.
Revenue at $4 per ticket:
Revenue = Number of tickets sold * Ticket price
Revenue = 400 tickets * $4
Revenue = $1600
Now, let's consider the manager's estimate that the ticket sales would decrease by 50 for each $1 increase in the ticket cost. This means that for every $1 increase in ticket price, 50 less tickets will be sold.
To find the new revenue at each price, we'll start by calculating the number of tickets sold for different price levels and then multiply that by the corresponding ticket price.
For a ticket price increase of $1, the number of tickets sold would decrease by 50. So:
Number of tickets sold = 400 - 50 = 350
Revenue at $5 per ticket:
Revenue = Number of tickets sold * Ticket price
Revenue = 350 tickets * $5
Revenue = $1750
For a ticket price increase of $2, the number of tickets sold would decrease by 50 for each $1, resulting in 400 - 2 * 50 = 300 tickets sold.
Revenue at $6 per ticket:
Revenue = Number of tickets sold * Ticket price
Revenue = 300 tickets * $6
Revenue = $1800
We can continue this process to calculate the revenue at different ticket prices and find the most profitable price to charge.
Ticket Price ($)| Number of tickets sold| Revenue ($)
-----------------|------------------------|---------------
$4 | 400 | $1600
$5 | 350 | $1750
$6 | 300 | $1800
$7 | 250 | $1750
$8 | 200 | $1600
From the table above, we can see that the most profitable ticket price is $6, as it yields the highest revenue of $1800.
Therefore, the most profitable price to charge is $6 per ticket.