Final answer:
The student's question involves finding the optimal ticket price for a theatre company to maximize revenue by considering a decrease in the number of subscribers based on price increases. By setting up a revenue function and using calculus to find its maximum, one can determine the best ticket pricing strategy.
Step-by-step explanation:
The student is asking about how to maximize revenue by adjusting the price of ticket sales for a theatre company. This question involves creating a revenue function based on the given parameters and finding its maximum value.
Maximizing Revenue
The initial number of subscribers is 500, and each ticket costs $270. For every $30 increase in price, 20 fewer people will renew their subscriptions, which indicates we are dealing with a downward-sloping demand curve. To find the optimal ticket price for maximum revenue, we can set up a revenue function R(p) = (500 - 20p) (270 + 30p), where p is the number of $30 increases. Differentiating this function with respect to p and finding the value of p for which the derivative is zero will give us the number of price increments to apply for revenue maximization. We would substitute the value of 'p' back into the ticket pricing formula to find the optimal ticket price. It's important to conduct a second derivative test or use another method to confirm that the calculated ticket price does indeed correspond to the maximum revenue point.