Chapter 7.3, Problem 20E

Business profits. New business owners typically experience a loss for a certain amount of time before the losses bottom out. Eventually the businesses show a profit. This pattern of profit and loss suggests a situation that could be modeled by a quadratic equation whose graph is a parabola opening up. Assume that you have begun a part-time business photographing weddings and that over the first three months of business you sustain losses of $\$200, \$130, and \$70$. (We can rephrase this by saying your cumulative profit at the end of month 1 was $-\$200,$ at the end of month 2 was $-\$330,$ and at the end of month 3 was $-\$400$.) Using quadratic regression, we can show that $P=30x^2-220x-10$ is the best quadratic equation that fits these data.

a. Verify that this equation fits the given data.

b. Using this model, at the end of what month do you expect to sustain your greatest cumulative loss? What is this loss?

c. At the end of what month do you expect to show your first cumulative profit? (By that, we mean that your total profits from the time the business started are greater than your total losses.)

d. What will your cumulative profit be by the end of month 10?

e. How much do you expect to earn during month 10?