Suppose we have opened a catfish hot dog stand at the mall which will be open 4 hours per day. We are test marketing our sales at various prices to attempt to determine the best price we should use for our catfish hot dogs. So, let: x = Number of y = Price of the hot dogs sold hot dogs (182, $1.50) (118, $3.00) (37, $4.00).

(a) Draw a scatterplot of these data values. Is it positively correlated, negatively correlated, or does it exhibit no correlation? What is the correlation coefficient for this model?

(b) What is the linear regression equation which fits this model? What is its slope? Give a verbal interpretation of the slope.

(c) Find the equation of the Revenue function R(x) for this model, and the equation of the Marginal Revenue function MR(x) for this function. Find the number of hot dogs sold that will maximize Revenue. What would be your daily revenue at this number of hot dogs? What is the price you should charge in order to generate this profit?

(d) Using the following Costs:

• $400 per month franchise fee

• $200 per month mall rental fee

• $0.60 per hot dog from the distributor

• $0.25 per roll from the distributor

• $9.50 per hour to pay the employee

Find the equation of the Cost function C(x) for this application. You should assume that there are 30 days for each month, and find the costs per day. What are the Fixed Costs? What are the Variable Costs?

(e) Find the equation of the Profit function P(x), and the equation of the Marginal Profit function MP(x).

(f) Find the number of hot dogs you would have to sell per day to in order to maximize your profit. What will be your daily profit at this number of hot dogs? What is the price you should charge in order to generate this profit?

(g) What is the Marginal Profit if you sell 160 catfish hot dogs per day? Interpret this number.

(h) Is your profit increasing or decreasing when you sell 100 catfish hot dogs? Explain how you determined this.

(i) Find all Break-Even points for this application.