**M****AT**** ****540 ****Quantitative**** Methods**

**Week 1** **Chapter 1**** Homework**

**2**. The Retread Tire Company recaps tires. The fixed annual cost of the recapping operation is $60,000. The variable cost of recapping a tire is $9. The company charges $25 to recap a tire.

a. For an annual volume of 12,000 tires, determine the total cost, total revenue, and profit.

**Formula is: Total cost = Fixed Cost + Total Variable Cost * Fixed Cost-60,000**

Total Variable Cost= 108,000 **(9**** x 12,000****) **

Total Cost= 168,000 (108,000+60,000**) **

Profits- 132,000 (300,000 -168,000**) **

Total Revenue- 300,000 **(12,000**** x 25****) **

Answer: The total cost is 168,000, the total revenue is 300**, 00****, and the profit ****is**** 132,000.**

b. Determine the annual break-even volume for the Retread Tire Company operation.

**Answer: The break even volume for Retread Tire Company operation is 3750 because you subtract 9-25, ****and then**** divide it by 60,000= 3750. So the break even volume is 3750.**

**4**. Evergreen Fertilizer Company produces fertilizer. The company’s fixed monthly cost is $25,000, and its variable cost per pound of fertilizer is $0.15. Evergreen sells the fertilizer for $0.40 per pound. Determine the monthly break-even volume for the company.

Fixed cost | 25,000 | ||||||
---|---|---|---|---|---|---|---|

Selling price per pound | 0.4 | ||||||

Variable cost per pound | 0.15 | ||||||

Contribution margin per pound | 0.25 | Selling price per pound – variable cost per pound | |||||

Breakeven volume | 100,000 | pounds | |||||

**12**. If Evergreen Fertilizer Company in Problem 4 changes the price of its fertilizer from $0.40 per pound to $0.60 per pound, what effects will the change has on the break-even volume?

Break Even Volume = Fixed cost / (Selling price per pound – Variable cost per pound)

**= 25000****/ (****0.60-0.15****) =**** 55,555.56 Pounds**

**The break even volume will reduce. **

**14**. If Evergreen Fertilizer Company increases its advertising expenditures by $14,000 per year, what effect will the increase have on the break-even volume computed in Problem 13?

Break Even Volume = Fixed cost / (Selling price per pound – Variable cost per pound)

= 14,000/(0.60-0.22) =3,6842.11 pounds

**20**. Annie McCoy, a student at Tech, plans to open a hot dog stand inside Tech’s football stadium during home games. There are seven home games scheduled for the upcoming season. She must pay the Tech athletic department a vendor’s fee of $3,000 for the season. Her stand and other equipment will cost her $4,500 for the season. She estimates that each hot dog she sells will cost her $0.35. She has talked to her friends at other universities who sell hot dogs at games. Based on their information and the athletic department’s forecast that each game will sell out, she anticipates that she will sell approximately 2,000 hot dogs during each game.

a. What price should she charge for a hot dog in order to break-even?

Total expenditure = 3000+4500=$7500

Since there are 7 games total number of hot dogs she can sell is approximately 2000*7 = 14000

Cost of each hot dog = 0.35

14000 hot dogs cost =0.35*14000=4900

Total cost for the season = 4900+7500 = 12400

She has to sell each hot dog at approximately = 12400/14000=0.885 ~ 0.89

b. What factors might occur during the season that would alter the volume sold and thus the break-even price Annie might charge?

Other vendors selling hot dogs

Games not getting sold out

Rain days

**22**. The College of business at Tech is planning to begin an online MBA program. The initial start-up cost for computing equipment, facilities, course development, and staff recruitment and development is $350,000. The college plans to charge tuition of $18,000 per student per year. However, the university administration will charge the college $12,000 per student for the first 100 students enrolled each year for administrative costs and its share of the tuition payments.

a. How many students does the college need to enroll in the first year to break-even?

The fixed cost is | 350,000 | |
---|---|---|

Selling Price | 18,000 | |

Variable cost | 12,000 | |

Contribution margin per student | 6,000 | |

Breakeven students | 58.33 | |

b. If the college can enroll 75 students the first year, how much profit will it make?

Number of students | 75 | ||||
---|---|---|---|---|---|

Total income | 1,350,000 | 75 students X 18,000 per student | |||

Variable cost | 900,000 | 75 students X 12,000 per student | |||

Fixed cost | 350,000 | ||||

Profit | $ 100,000 | Total income – variable cost – fixed cost |

c. The college believes it can increase tuition to $24,000, but doing so would reduce enrollment to 35. Should the college consider doing so?

Number of students | 35 | |||
---|---|---|---|---|

Total income | 840,000 | 35 students X 24,000 per student | ||

Variable cost | 420,000 | 35 students X 12,000 per student | ||

Fixed cost | 350,000 | |||

Profit | 70,000 | |||

**Week 1**** ****Chapter 11 Homework**

**18**. The following probabilities for grades in management science have been determined based on past records:

Grades | Probability |

A | .10 |

B | .30 |

C | .40 |

D | .10 |

F | .10 |

1.00 |

The grades are assigned on a 4.0 scale, where an A is a 4.0, B a 3.0, and so on. Determine the expected grade and variance for the course

Answer: The expected GPA is 2.2

**20.** An investment firm is considering two alternative investments, A and B, under two possible future sets of economic conditions, good and poor, There is a .60 probability of good economic conditions occurring and a .40 probability of poor economic conditions occurring. The expected gains and losses under each economic type of conditions are shown in the following table:

Economic Conditions

Investments | Good | Poor |

A | $900,000 | -$800,000 |

B | $120,000 | 70,000 |

Using the expected value of each investment alternative, determine which should be selected.

**Expected value of investment A = Probability of economic condition as good * payoff when economic condition is good + Probability of economic condition as Poor * payoff when economic condition is poor**

**=900000*0.6****+ (****-800000)*0.4 =$220,000 **

**Expected value of investment B = Probability of economic condition as good * payoff when economic condition is good + Probability of economic condition as Poor * payoff when economic condition is poor**

**=120000*0.6+70000*0.4 =$100,000 **

**The expected payoff of Investment A is higher so investment A should be selected.**

**26**. The weight of bags of fertilizer is normally distributed, with a mean of 50 pounds and a standard deviation of 6 pounds. What is the probability that a bag of fertilizer will weigh between 45 and 55 pounds?

P[45 < X < 55 ] = P[(45-50)/6 < Z < (55-50)/6] = P[-0.83 < Z < 0.83 ] =

2*P [0< Z < 0.83] = 0.2967 = **0.5934**

**28**. The Polo Development Firm is building a shopping center. It has informed renters that their rental spaces will be ready to occupy in 19 months. If the expected time until the shopping center is completed is estimated to be 14 months, with a standard deviation of 4 months, what is the probability that the renters will not be able to occupy in 19 months?

**Assume normal distribution**

**We need to calculate P(X>19)**

**Given mean = 14**

**Standard Deviation = 4**

**P(X>19) = ****P (****z****> (****19-14)/4) **

**=****P (****z>1.25)**

**From table the value of probability is 0.1056**

**T****he probability that the renters will not be able to move in 19 months is 10.56%. **

**30.** The manager of the local National Video Store sells videocassette recorders at discount prices. If the store does not have a video recorder in stock when customers want to buy one, it will lose the sale because the customer will purchase a recorder from one of the many local competitors. The problem is that the cost of renting warehouse space to keep enough recorders in inventory to meet all demand is excessively high. The manager has determined that if 90% of customer demand for recorders can be met, them the combined cost of lost sales and inventory will be minimized. The manager has estimated that monthly demand for recorders s normally distributed, with a mean of 180 recorders and a standard deviation of 60. Determine the number of recorders the manager should order each month to meet 90% of customer demand.

The demand for recorders is normally distributed with and STD deviation,

P (demand<=180) =0.9 can be given by z=1.28 and,

**To**** meet 90% of customer demand, the manager should order 257 recorders every month.**