Final answer:
To generate $182,600 in total revenue, the theater must sell 4100 tickets at $26 and 1900 tickets at $40. Raising ticket prices can affect the overall revenue due to the price elasticity of demand and the associated downward-sloping demand curve.
Step-by-step explanation:
The student's question pertains to a math problem involving a 6000-seat theater with tickets priced at $26 and $40, aiming for a sellout performance to generate $182,600 in total revenue. We know that the number of $26 tickets that should be sold is given as 4100. Let's represent the number of $40 tickets that need to be sold as x. Thus, we can set up the following equation to represent the total revenue (TR) from ticket sales:
TR = ($26 × 4100) + ($40 × x)
Plugging in the known values and the total revenue target:
$182,600 = ($26 × 4100) + ($40 × x)
With simple algebra, we solve for x:
$182,600 = $106,600 + $40x
$182,600 - $106,600 = $40x
$76,000 = $40x
x = $76,000 / $40
x = 1900
Therefore, in order for the theater to sell out and generate a total revenue of $182,600, they need to sell 4100 tickets at $26 and 1900 tickets at $40.
Considering the question of whether raising prices brings in more revenue, it's important to understand the price elasticity of demand. In the case of fixed costs, and assuming the demand curve is downward sloping, raising prices would lead to fewer tickets sold. Hence, the optimal strategy for maximizing revenue depends on the price sensitivity (elasticity) of the customers.