Final answer:
The task involves calculating the break-even point and maximizing profit for a business. The profit function, p(x) = 1300x - 161x², can help find these values, using principles from economics and calculus.
Step-by-step explanation:
The profit function given by a student is p(x) = 1300x - 161x², where p(x) represents profit and x represents the number of units sold. To find the break-even point, we need to set the profit function equal to zero and solve for x. However, we're also tasked with finding the price that will maximize profit. This is a classic problem of maximizing revenue and profit in a business context.
To find the maximum revenue, we can look at the provided example where a firm has chosen a price and quantity and calculated the total revenue, total cost, and profit. For instance, at a quantity of 40, with a price of $16 per unit, the firm's total revenue is $640 and the total cost is $580, resulting in profits of $60. The firm doesn't necessarily make a profit at every level of output, as shown in one example where total costs exceed revenues from 0 to approximately 30 units, leading to losses.
For finding the price that will maximize profit, we can use the fact that profit is maximized where marginal revenue equals marginal cost. This is not directly given in the student's function, but the principle could be used in general to find the maximum profit point. Another way is to use calculus to find the derivative of the profit function and set it equal to zero to find the critical points, and then determine which point corresponds to the maximum profit.