To determine the profit-maximizing quantity of tickets, we need to follow a process involving marginal revenue (MR) and marginal cost (MC).

Step 1: Write the Demand Function

The inverse demand function is given as:

$P = 120 - 0.005Q$

Here, $P$ is the price and $Q$ is the quantity of tickets.

Step 2: Find the Total Revenue (TR) Function

Total revenue is calculated as price times quantity:

$TR = P \times Q = (120 - 0.005Q) \times Q = 120Q - 0.005Q^2$

Step 3: Find the Marginal Revenue (MR) Function

Marginal revenue is the derivative of total revenue with respect to $Q$:

$MR = \frac{d(TR)}{dQ} = 120 - 0.01Q$

Step 4: Set Marginal Revenue Equal to Marginal Cost

To maximize profits, we set marginal revenue equal to marginal cost (MC). The marginal cost is constant at $5:

$MR = MC$

$120 - 0.01Q = 5$

Step 5: Solve for $Q$

Now, solve for $Q$:

$120 - 5 = 0.01Q$

$115 = 0.01Q$

$Q = \frac{115}{0.01} = 11,500$

Conclusion:

The theater should sell 11,500 tickets to maximize its profit.

Let me know if you want details or have any questions. Here are 5 related questions for further exploration:

1. How would the profit-maximizing quantity change if marginal costs were higher?
2. What happens to profit-maximizing quantity if demand becomes more elastic?
3. How does consumer surplus change at 11,500 tickets?
4. How would profits change if ticket prices were capped at $100?
5. What is the total revenue when 11,500 tickets are sold?

Tip: In economics, the marginal revenue is often lower than price due to the decreasing demand curve.