1. For the following total revenue and total cost functions of a firm:
   TR = 802.5Q – 10Q²
   TC = (2/3)Q³ -30Q² + 672Q +4000
   

1. Determine the level of output at which the firm maximizes total profit
2. Calculate the profit
   (20 points)

• Reference problem on pp. 104-106 and graphs on p. 77.

1. Solution writes a total profit equation (TR equation minus the TC equation and simplify) and takes its derivative to get a marginal profit equation.
2. If you set the marginal profit or derivative equation equal to zero you find the points where the rate of change is zero; i.e., the peak or trough (see diagram of profit function in text).
3. To solve it you will probably want to use the Quadratic Formula. If you are unfamiliar with the Quadratic see http://www.purplemath.com/modules/quadform.htm
4. The quadratic formula gives two values of Q for which the marginal profit is zero. In the text example, they take the second derivative (the derivative of the marginal profit function) and plug in each quantity. If the 2^nd derivative is negative you have reached a peak, if negative, a trough. You can take the 2^nd derivative if you prefer but for the functions you will encounter in this course, the answer is ALWAYS THE LARGER ONE. Note that on the total profit graph (p. 77) there is both a peak and a trough. The peak occurs at the larger quantity.

Candice operates an ice cream parlor in a small town in Tristate area. She knows that this a monopolistically competitive business because other producers in the area supply different flavors of ice cream. Candice runs her business as efficiently as possible, to maximize her profits. This year, Candice charges $5 per ice cream and experiences marginal cost of $3 and average total cost of $4 per ice cream at the optimal level of output. Does Candice have profits in short term (this year)? Can you predict how ice cream market conditions for Candice will be changing in the near future? What will be Candice's profit in the long-run?

Cedar Point amusement park management is preparing the park's annual promotional plan for the coming season. Several advertising alternatives exist: newspaper, television, radio, and displays at recreational shows. The information below shows the characteristics associated with each of the advertising alternatives, as well as the maximum number of placements available in each medium. Given an advertising budget of $125,000, how many placements should be made in each medium to maximize total audience exposure? Formulate this as a linear programming problem to be solved using QM.

A 600​-room hotel can rent every one of its rooms at $90 per room. For each​ $1 increase in​ rent,

3 fewer rooms are rented. Each rented room costs the hotel​ $10 to service per day. How much should the hotel charge for each room to maximize its daily​ profit? What is the maximum daily​ profit?

Science has been recognized as a critical input in good decision making for park managers. How does the use of science benefit park management decisions?

In January 2017, Mitzu Co. pays $2,600,000 for a tract of land with two buildings on it. It plans to demolish Building 1 and build a new store in its place. Building 2 will be a company office; it is appraised at $750,000, with a useful life of 20 years and a $85,000 salvage value. A lighted parking lot near Building 1 has improvements (Land Improvements 1) valued at $360,000 that are expected to last another 12 years with no salvage value. Without the buildings and improvements, the tract of land is valued at $1,890,000. The company also incurs the following additional costs:

Cost to demolish Building 1 $ 339,400
Cost of additional land grading 191,400
Cost to construct new building (Building 3), having a useful life of 25 years and a $400,000 salvage value 2,302,000
Cost of new land improvements (Land Improvements 2) near Building 2 having a 20-year useful life and no salvage value 168,000

Allocate the costs incurred by Mitzu to the appropriate columns and total each column.

2. Prepare a single journal entry to record all the incurred costs assuming they are paid in cash on January 1, 2017.

3. Using the straight-line method, prepare the December 31 adjusting entries to record depreciation for the 12 months of 2017 when these assets were in use.

You are the revenue manager of a 200-room hotel in Memphis. The management controllable costs (variable room costs) incurred when selling 1 room are $50. You are interested in evaluating hotel performance for two scenarios: fixed pricing and differential pricing.

Scenario one is that you used a fixed pricing strategy: at a selling pricing $200 per night your hotel would sell 150 rooms on a given day.

Scenario two is that you implement a three-price strategy: low rate $150 per night, regular rate $200 per night, and high rate $250 per night. Your hotel would sell 100 low-priced rooms, 60 regular-priced rooms, and 20 high-priced rooms per day.

2. Discuss which scenario generates more revenue for your hotel and why?

4. Leslie has been tasked with putting together a statistical report regarding the use of a particular park in her hometown. Previous data shows that 72% of the residents living in the town visited the park in the last month. She wants to know the probability that more than 98 individuals in a random sample of 150 residents have visited the park in the last month. Please Answer A-C

(a) Based on the information given, is the sample proportion p̂ of residents who have visited the park in the last month approximately normally distributed? Check the appropriate conditions to justify your answer

(b) What is the sample proportion p̂ for Leslies sample of 150 residents? Round your answer to two decimal places?

(c) What is the probability that more than 98 individuals in the random sample of 150 residents have visited the park in the last month?