Question

2-6% each part 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: Price of the Number of y hot dogs $1.50 $3.00 $4.00 hot dogs sold 182 118 39 Notice that we have put these variables in the same order that we did in Section 12, uhich is different (a) Draw a scatterplot of these data values. Is it positively correlated, negatively correlated, or does (b) What is the linear regression equation which fits this model? What is its slope? Give a verbal from what we have done earlier in the course. it exhibit no correlation? What is the correlation coefficient for this model? 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(r) 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 e $200 per month mall rental fee . $0.60 per hot dog from the distributor . $0.25 per roll from the distributor $9.75 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(ar), and the equation of the Marginal Profit function MP() (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? (8) 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.

Answer 1

a) scatter plot :

5- 240 200 160 120 80 48

According to the scatter plot the data is negatively related that is if price increase then the demand of the hot dog decrease and vice versa.

The correlation coefficiency is

			$x^{2}$	$y^{2}$
182	1.5	273	33124	2.25
118	3	354	13924	9
39	4	156	1521	16
339	8.5	783	48,569	27.25

783 -1/3 339 8.5 V27.25- 1/3* (8.5)2 y48569 - 1/3 * (339)2

r = -0.9846

b) Regression Equation = ŷ = bX + a

b = SP/SS_X

a = M_Y - bM_X

X - Mx	Y - My	(X - Mx)2	(X - Mx)(Y - My)
69	-1.3333	4761	-92
5	0.1667	25	0.833
-74	1.1667	5476	-86.3333

Sum of X = 339
Sum of Y = 8.5
Mean X = 113
Mean Y = 2.8333
Sum of squares (SS_X) = 10262
Sum of products (SP) = -177.5

b = SP/SS_X = -177.5/10262 = -0.0173

a = M_Y - bM_X = 2.83 - (-0.02*113) = 4.78787

y = -0.0173X + 4.78787

As the slope of this linear regression equation is negative, therefore the number of hot dog sold is inverly related with the price of the hot dog. Therefore if price of hot dog increses then demand of the hot dog decreases and vice versa.

c) The total revenue function is:

TR = (-0.0173X + 4.78787)x

= -0.0173x²+ 4.78787x

The marginal revenue is :

MR = dTR/dx = -0.0346x + 4.7878

-0.0346x + 4.7878 = 0

x = 138.37

Therefore the total revenue at this level of hot dog sold is

TR = -0.0173x²+ 4.78787x = -0.0173(138.37)²+ 4.7878*138.37

= 331.26

In order to generate the total revenue , the price is

y = -0.0173( 138.37)+ 4.78787

= 2.394

d) Cost function = (Input price) x

Therefore the profit of the franchse is (per month)

$\pi =TR-TC$

= 331.26 *30 - 400

= 9537.8

Therefore total profit in a month is $9537.8

e) The profit function is total revenue minus total cost

P(z) = TR(r)-TC(z)

= -0.0173x²+ 4.78787x - (Input price) x

marginal revenue = -0.0346x + 4.7878

Marginal cost is the derivative of the cost function = input value

MP(z) = MR(x)-MC(z)