Question

4) The profit that a company earns in dollars is given by the function P(q) = -2q2 + 100q - 300 where q is the number of items that the company sells IN THOUSDANDS. Suppose that this company is ONLY capable of producing and selling 10 thousand items or fewer. (a) (4 points) Based on this mathematical model, what are the number of items this company needs to sell in order to break even? ROUND YOUR VALUE to the second decimal place. Explain/show how you solved the problem: Write a complete, content-rich sentence to explain the answer. (b) (4 points) Remember that the company cannot sell a fraction of an item. For what values of q items are sold at a loss (also known as negative profit)? Which items are sold at a profit? Show your answers with appropriate notation (<s, >, 2, 5, 7) as needed. Write a complete, content-rich sentences to explain the answer.

Answer 1

We are given profit function of a firm and asked to find break even point for the firm and values of output for which firm incurs loss

P(q) = -29% + 100q - 300 where qis output(in thousands)

Part (a)

We are asked quantity that must be sold to break even, that is the point where profits are 0 (costs are equal to revenue)

10000 – 2400 -4 P(q) = –20 + 100q - 300 = 0 = -29% + 100q – 300 = 0 finding roots of this quadratic equation -100 V1002 – 4*(-2)*(-300) -100 2*(-2) -100 = V7600_-100 = 87.17 -100 – 87.17 -187.17 either q=2 = 46.79 -4 -100 + 87.17 - = 3.20 or q= 2 -4

It is mentioned that firm can produce only 10,000 thousand or fewer, that is output q cannot be greater than 10

:. we take the solution q = 3.2

Hence firm must produce and sell 3200 (3.2*1000) items to break even

Part (b)

We are asked to find the quantity for which firms incurs loss and profits

Since we have found that at value of output 3.2, firm is break even that is profit are 0 :. for sale of output less than 3.2(3200), firm incurs a loss for sale of output more than 3.2(3200), firm incurs a profit P(q) > 0 when q> 3.2 P(q) < 0 when q<3.2