When a theater charges $50 per ticket, a sold-out crowd of 600 will buy tickets. For every $5 increase in ticket price, 6 fewer people buy tickets. What ticket price would maximize the theater’s revenue? (Recall that revenue = price • quantity).
r = price * quantity
= p(600-((p-50)/5)*6)
= 660p - 6/5 p^2
max r at p = 275
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To maximize the theater's revenue, we need to determine the ticket price that will result in the highest total revenue.
Let's start by calculating the revenue when the ticket price is $50, with a sold-out crowd of 600 people:
Revenue at $50 = price * quantity = $50 * 600 = $30,000
Now, we need to find out how the quantity changes as the ticket price increases. According to the given information, for every $5 increase in ticket price, 6 fewer people buy tickets.
Let's represent the change in quantity as ΔQ and the change in price as ΔP.
ΔQ = -6 (for every $5 increase in price)
To find the change in price required for a specific change in quantity, we can use the equation:
ΔP = (ΔQ / -6) * $5
To maximize revenue, we need to find the value of ΔP that would result in a ΔQ of 600 (the entire crowd). Then we can add that change in price to the original ticket price of $50 to get the optimal ticket price.
ΔQ = 600
ΔP = (600 / -6) * $5 = -100 * $5 = -$500
Since the value of ΔP is negative, we need to subtract it from the original ticket price to find the optimal ticket price:
Optimal ticket price = $50 - $500 = -$450
However, a negative ticket price doesn't make sense in this context. Thus, the theater cannot maximize its revenue by increasing the ticket price.
Therefore, the ticket price that would maximize the theater's revenue is $50.
To maximize the theater's revenue, we need to find the ticket price that will result in the highest total revenue.
Let's break down the problem step by step:
1. Start with the given information: when the ticket price is $50, the theater sells out with a crowd of 600 people.
2. Determine the relationship between the ticket price and the number of people buying tickets. From the information provided, we know that for every $5 increase in ticket price, 6 fewer people buy tickets. This means that the rate of change is 6 people per $5 increase in price.
3. Calculate the number of people buying tickets at different prices. We can start with the original ticket price of $50 and increase it by $5 increments, while decreasing the number of people accordingly. Below is a table illustrating this:
Ticket Price | Number of People Buying Tickets
-----------------------------------------------
$50 600
$55 594
$60 588
$65 582
... ...
Continue this pattern until you notice a change.
4. Calculate the total revenue at each ticket price by multiplying the ticket price by the number of people buying tickets. Continuing with the example above:
Ticket Price | Number of People Buying Tickets | Revenue
--------------------------------------------------------------------------
$50 600 $30,000
$55 594 $32,670
$60 588 $35,280
$65 582 $37,830
... ... ...
5. Identify the ticket price that generates the maximum revenue. Look for the highest revenue value obtained from the table.
In this case, you should continue the process until the revenue begins to decrease.