Final answer:
a. The company's profits or losses are $30159 at a quantity of 5 units and a price of $25 per unit. b. By comparing the average cost to the price per unit, we can tell whether the company is making or losing money. c. The marginal unit produced is not adding to profits.
Step-by-step explanation:
a. To determine the company's profits or losses, we need to calculate the total revenue and the total cost. Total revenue can be calculated by multiplying the quantity of units (5) by the price per unit ($25), which equals $125. Total cost can be calculated by using the profit function: P(x) = -53x^2 + 6461x - 696. Substituting x = 5 into the profit function gives us P(5) = -53(5)^2 + 6461(5) - 696 = -1325 + 32305 - 696 = 30284. Therefore, the company's profits or losses are $30284 - $125 = $30159.
b. At a glance, we can determine whether the company is making or losing money at this price by looking at the average cost. Average cost can be calculated by dividing the total cost by the quantity of units. In this case, the average cost is $30284/5 = $6056. Since the average cost is higher than the price per unit ($25), the company is losing money.
c. The marginal unit refers to the additional unit produced. To determine whether the marginal unit is adding to profits, we need to compare the marginal cost to the marginal revenue. The marginal cost can be calculated by taking the derivative of the profit function and substituting x = 5: P'(x) = -106x + 6461. P'(5) = -106(5) + 6461 = -530 + 6461 = 5931. Thus, the marginal cost when producing the 6th unit would be $5931. The marginal revenue can be calculated by subtracting the price per unit from the total revenue: Marginal revenue = Total revenue - (Quantity x Price per unit) = $125 - (5 x $25) = $0. Since the marginal cost ($5931) is greater than the marginal revenue ($0), the marginal unit produced is not adding to profits.