# 1.7: Problems

- Page ID
- 82021

*(Adapted from Glover, Lunsford, and Fleming)*

There are 20 chickens and 2 foxes present on an acre of land on the morning of June 1, 1990. During the day, half of the chickens leave the acre of land to search for food, but both of the foxes stay on the acre of land. By nightfall, all of the chickens that left have returned. At the end of the day, the total number of chickens is 15 and the total number of foxes is 2. Using the accounting principle, explain what happened. Remember to explicitly define the counted property, the system and its boundary, and the time period. Also state any assumptions you made in order to solve the problem.

*(Adapted from Glover, Lunsford, and Fleming)*

A baseball stadium can hold a maximum of 55,000 spectators. Before the gates open, there are only 200 stadium personnel in the stadium. When the gates open, there are 500 times more spectators than media personnel that enter the stadium. There are 50 baseball players that enter the stadium. During the game, one-fourth of the spectators leave. There are 30,000 spectators at the end of the game. No players, stadium personnel, or media personnel leave during the game. Using the accounting principle, calculate the total number of people in the stadium at the end of the game. [Hint: Try making a table to solve this problem]

Billy has a job delivering newspapers every morning before school. Billy is industrious and actually works for two different newspapers, the *Herald* and the *Post*. Billy's route covers five different streets. Every house gets either the *Herald* or the *Post* newspaper, but not both. Each morning Billy receives 140 newspapers from the *Herald* and 190 newspapers from the *Post*. The available data for the number of houses and newspapers delivered is given in the following table:

Streets |
Total Number of Houses |
Herald Houses |
Post Houses |
---|---|---|---|

Elm | 64 | 34 | \(?\) |

Park | 58 | 27 | \(?\) |

Oak | 37 | \(?\) | 20 |

Main | 84 | 35 | \(?\) |

1st | 75 | \(?\) | 50 |

Select a suitable system (or systems) and use the accounting principle to determine the total number of newspapers, including how many *Heralds* and *Posts*, Billy accumulates each day. (You will find that setting up a table will help you solve this problem.) Remember to explicitly define the counted property, the system and its boundary, and the time period. Also state any assumptions you made in order to solve the problem.

Financial data on Mr. Jones are presented in the table below:

Transaction |
January |
February |
March |
April |
May |
June |
---|---|---|---|---|---|---|

Deposit | $2,600 | $2,300 | $2,000 | $2,100 | $2,400 | $2,600 |

Expenses | ||||||

Mortgage | 1,000 | 1,000 | 1,000 | 1,000 | 1,000 | 1,000 |

Auto | 500 | 500 | 5000 | 500 | 500 | 500 |

Bills | 1,000 | 700 | 1,100 | 900 | 400 | 800 |

Food | 190 | 160 | 210 | 180 | 200 | 140 |

Insurance | 600 | 0 | 0 | 0 | 0 | 0 |

The initial balance at the beginning of January was $5,000. You have been asked to evaluate his financial health by tracking the monthly balance in his checking account. Use a conservation and accounting framework to solve the problem. [Hint: Set up a spreadsheet showing transports of money in and out of the account and the change in the account. Treat the account as a system.] Consider the following two cases and answer the questions:

**Case A --- A non-interest bearing checking account.**

Develop a spreadsheet that shows the balance in Mr. Jones' account at the end of each month. Plot the balance in the account as a function of time.

- Is he ever bankrupt? If so, in what month does this occur? If not, what is the minimum positive balance and when does it occur?
- What is the net change in the amount of money in the checking account for this six-month period?
- If there is to be no net change in the amount of money in the account over the six months, what amount, on average, must Mr. Jones deposit monthly to his account? Does this answer depend on the initial balance in the account?

**Case B --- An interest bearing checking account.**

If at the end of each month, the money added to the amount in the form of interest is given by the following equation

\[ \text{Dollars Added} = P \cdot i \nonumber \]

where: \(P\) = the *average* amount of money in the account during the month

\(i = 0.005 ($ \text{ interest})/($ \text{ of principal})\) [Equivalent of 6% annually]

Calculate the balance at the end of each month and plot the balance as a function of time. [Should you treat the interest dollars as money produced or money transported? Does it really make any difference?]

- Is he ever bankrupt? If so, in what month does this occur? If not, what is the minimum positive balance and when does it occur?
- What is the net change in the amount of money in the checking account for this six-month period?
- If there is to be no net change in the amount of money in the account over the six months, what amount, on average, must Mr. Jones deposit monthly to his account? Does this answer depend on the initial balance in the account? Compare your answer to that is Case A. Does it make sense?

The Food Lion grocery store in Corolla, NC is located on the Outer Banks and serves a local population of residents and tourists of approximately 5,000 people. Only 20% of the people are permanent residents of the Outer Banks. The remaining 80% of the people – the tourists – rent cottages on a weekly basis and stay only one week. One-half of the tourists arrive on Saturday and the rest arrive on Sunday. As manager for the Food Lion store, you must order supplies on a weekly basis. Deliveries are made twice a week on Tuesday and Friday.

A typical family uses the following groceries in a week:

Used per Family (4 persons) | |
---|---|

Produce | \(14 \text{ lbs}\) |

Paper Products | \(1.5 \text{ ft}^3\) |

Milk | \(2 \text{ gallons}\) |

Beverages | \(56 \text{ cans}\) |

The residents do all of their shopping for the week on Wednesday to avoid the weekend crowds. The tourists buy 80% of their groceries on the day after they arrive and buy the remaining groceries four days after they arrive. Using the accounting principle, answer the following questions.

- Develop a table and/or graph showing the daily demand for the groceries starting with Sunday.
- What day has the peak demand?
- If you are required to have a minimum of 10% reserve of your average daily sales, how much storage space would you need for paper products?

- Determine the minimum amount of groceries that must be delivered on Tuesday and Friday to meet customer demand during the summer tourist season.
- Your suppliers would like to increase the number of deliveries per week; however, they would also like to deliver
*approximately*the same amount on each trip. How many deliveries per week would you suggest, what days of the week should they be made on, and how much should be delivered on each trip?

Fuel oil is used to fire a small power plant to generate electricity. The fuel oil fires a boiler that generates steam which in turn drives a steam turbine connected to an electric generator. The power plant requires \(160 \text{ lbm/h}\) of fuel oil to steadily generate electrical energy at the rate of \(290 \text{ kW}\). In addition, the combustion of the fuel requires \(15.8 \text{ lbm}\) of air for each pound-mass of fuel supplied to the power plant.

Figure \(\PageIndex{1}\): Inputs and outputs of the power plant.

Use the accounting concept, clearly stating all assumptions, and determine:

- the rate at which air must be supplied to the power plant and the rate at which exhaust and ash are produced, in \(\text{lbm/h}\).
- the amount of fuel required and the amount of exhaust and ash produced, in \(\text{lbm}\), if the plant operates continuously for 24 hours.

GitErDone, Inc. (GED), the premier manufacturer of Indiana widgets, manufactures RH Widgets. The GED factory consists of three areas: a repair shop (RS), a production shop (PS), and an inspection and delivery area (IDA). Over a five-day period, the staff recorded the following operating information:

Day | Damaged widgets returned to RS from customers | Widgets produced in PS and sent to IDA | Widgets rejected in IDA and returned to RS for repair | Good-as-new widgets sent from RS to IDA | Widgets delivered by IDA to customers |
---|---|---|---|---|---|

1 | 33 | 13,600 | 1362 | 1350 | 14,000 |

2 | 52 | 12,600 | 1258 | 1150 | 14,000 |

3 | 47 | 14,600 | 1465 | 1050 | 14,000 |

4 | 28 | 14,000 | 1395 | 1250 | 14,000 |

5 | 40 | 13,200 | 1320 | 1350 | 14,000 |

At the beginning of Day 1, 500 damaged widgets were in the repair shop and 21000 widgets were in the inspection and delivery area. No widgets are stored in the production shop. The storage area in the IDA is small and can hold only a five-day delivery inventory.

Apply the generic accounting principle to an appropriate system (the factory, RS, PS, and/or IDA) to answer the following questions. Although hand calculations are acceptable, a spreadsheet helps is ideal for setting up this problem. If you use a spreadsheet, include a copy with your solution.

- Determine the inventory of widgets and damaged widgets and their location at the end of each day. Show both tabular and graphical results.
- If the factory inventory of widgets is increasing, determine the number of days to fill the available widget storage area. If the factory inventory of widgets is decreasing, determine the number of days before the inventory of widgets drops to zero. [Hint: Use the average daily values.]
- Under optimum operating conditions, the average inventory of widgets in the factory is constant. Using the given operating data, estimate how many widgets on-average must be delivered to customers daily to achieve optimum conditions. Under these conditions, what is happening to the inventory of damaged widgets in the repair shop? Is this a problem?