## Planning

Problems and tasks from a variety of sources intended to illustrate the way mathematics arises in life and work.

Setting Prices. Find the optimal price for a student magazine given polling information on the number that will be purchased at different prices.

Mixing Fertilizer. In preparing fertilizer for a garden, a homeowner poured one quart of concentrated liquid fertilizer into a two-gallon can, and filled the can with water. She then discovered that the proper ratio of fertilizer to water should have been 1:3. How much more liquid fertilizer should she add to the current mixture to obtain the desired concentration?

Critical Path Analysis. Make a critical path diagram for planning a dinner party.

Routing Elevators. Determine the time it takes for elevators in an office building to transport workers to their offices if they all arrive just before 9:00 a.m. Would it help if certain elevators were restricted to serving specific floors?

Repairing Lights. An electrician is called on to repair an old fixture with three fluorescent lights. Each light has a tube, a ballast, and two wires, one or more of which may be the source of the problem. Because of the way the fixture is wired, any bad ballast will cause the entire fixture to fail, whereas a bad wire or bad tube will just cause one light to fail. Devise a strategy for diagnosing the problem, bearing in mind the possibility that there may be more than one bad component.

Planning a Vacation. A young couple with two children and an eight-year-old car is trying to decide whether to fly or drive 1500 miles each way for a vacation to Disney World. They have saved \$2000 for the trip and estimate that four sale-priced airline tickets would take about half that. If they drive, they will have to pay for travel expenses (motels, food, gas, and tolls), will lose wages for extra days off from work, and will run the risk of a breakdown with their eight-year old car. What should they do?

Roasting Coffee. A production manager at a coffee roasting facility keeps track of data on the projected and actual demand for various blends of coffee. Since the projected demand often exceeds the facility's total capacity, he must prioritize his production schedule in the hope of meeting real demand. Given the data for the past twelve months, and the projected demand for the next two, what production schedule should he order for those two months?

Building a Hospital. A city council is looking into the wisdom of building a hospital on land that the city has recently acquired in a rapidly growing area. In addition to the cost-benefit issues involved in running the hospital, there are matters of additional infrastructure (roads, sewer, waste disposal), jobs (first for construction, then for running the hospital), housing (both for hospital employees and for those attracted because of the increased health services), and taxes (as a non-profit institution, the hospital would be exempt from most taxes). Each of these factors changes with time--some relatively predictably, others less so (e.g., rates of population growth). A mathematical model can link all these variables to produce a variety of "what if" scenarios. It is not uncommon for such models to reveal patterns that cannot be foreseen when the variables are considered one at a time.