Solution:

To maximize profit, a firm needs to determine the output level at which the marginal revenue (MR) is equal to the marginal cost (MC). This is known as the profit-maximizing output level.

The given information:

Total revenue (TR) = 500Q - 5Q^2
Total cost (TC) = 200 + 50Q
Where Q is the quantity produced.

To find the profit-maximizing output level, we need to follow the below steps:

Step 1: Find the marginal revenue (MR) function by taking the derivative of total revenue with respect to quantity.

MR = d(TR)/d(Q) = 500 - 10Q

Step 2: Find the marginal cost (MC) function by taking the derivative of total cost with respect to quantity.

MC = d(TC)/d(Q) = 50

Step 3: Set MR equal to MC and solve for Q.

500 - 10Q = 50
450 = 10Q
Q = 45

Therefore, the profit-maximizing output level is 45 units.

Conclusion:

To maximize profit, the firm should produce 45 units. At this level of production, the marginal revenue is equal to the marginal cost, and the firm will earn maximum profits.