You are working temporarily for the Department of Health and Environmental Control. The director is concerned about the issue of teenage pregnancy in their reg
ion. You have decided that your team will analyze the situation and determine if it is really a problem in this region. You gather the following 2000 data.
Age 10-14 Pregnant
Age 15-17 Pregnant 350 303 422 201 156 523 263 330 123 467 421 179
Age 18- 19 Pregnant 571 567 691 356 357 970 434 427 221 950 713 311
Age 10-14 births 17 13 29 18 11 33 9 16 10 24 18 8
Age 15-17 births 281 206 307 184 109 442 201 256 113 446 343 145
Age 18-19 births 437 466 546 326 254 803 345 444 199 686 615 261
births-unmarried births-unmarried births-unmarried 16 13 28 15 10 32 7 14 9 22 15 7 164 151 251 137 99 293 113 160 78 279 219 114 193 233 366 180 161 396 168 210 106 331 328 162
1 2 3 4 5 6 7 8 9 10 11 12
29 24 40 21 16 44 17 23 13 41 28 9
1998 Age 10-14 15-17 18-19 Pregnancies 320 4041 6387 1999 Age 10-14 15-17 Pregnancies 309 3882 Births 208 3048 Births 231 3222 5164
Build a mathematical model and use it to determine if there is a problem or not. Prepare an article for the newspaper that highlights your result in a novel mathematical relationship or comparison that will capture the attention of the youth.
A South Sea Island Resort
A south sea island chain has decided to transform one of their islands into a resort. This roughly circular island, about 5 kilometers across, contains a mountain that covers the entire island. The mountain is approximately conical, is about 1000 meters high at the center, appears to be sandy, and has little vegetation on it. It has been proposed to lease some fire-fighting ships and wash the mountain into the harbor. It is desired to accomplish this as quickly as possible. Build a mathematical model for washing away the mountain. Use your model to respond to the questions below.
? ? ? ?
How should the stream of water be directed at the mountain, as a function of time? How long will it take using a single fire-fighting ship? Could the use of 2 (or 3, 4, etc.) fire-fighting ships decrease the time by more than a factor of 2 (or 3, 4, etc.)? Make a recommendation to the resort committee about what do.
The First National Bank has just been robbed (the position of the bank on the map is marked). The clerk pressed the silent alarm to the police station. The police immediately sent out police cars to establish road blocks at the major street junctions leading out of town. Additionally, 2 police cars were dispatched to the bank. See the attached map. The Bank is located at the corner of 8th Ave. and Colorado Blvd. and is marked with the letter B. The main exits where the two road blocks are set up are at the intersection of Interstate 70 and Colorado Blvd, and Interstate 70 (past Riverside Drive). These are marked with a RB1 and RB2 symbol.
Assume the robbers left the bank just before the police cars arrived. Develop an efficient algorithm for the police cars to sweep the area in order to force the bank robbers (who were fleeing by car) into one of the established road blocks.
Assume that no cars break down during the chase or run out of gas. Further assume that the robbers do not decide to flee via other transportation means.
Design of an Airline Terminal
The design of airline terminals varies widely. The sketches below show airline terminals from several cities. The designs are quite dissimilar. Some involve circular arcs; others are rectangular; some are quite irregular. Which is optimal for operations? Develop a mathematical model for airport design and operation. Use your model to argue for the optimality of your specified design. Explain how it would operate.
Boston-Logan International Munich International
Ronald Reagan Washington National
Your team has been approached by the Forest Service to help allocate resources to fight wildfires. In particular, the Forest Service is concerned about wildfires in a wilderness area consisting of small trees and brush in a park shaped like a square with dimensions 80 km on a side. Several years ago, the Forest Service constructed a network of north-south and east-west firebreaks that form a rectangular grid across the interior of the entire wilderness area. The firebreaks were built at 5 km intervals. Wildfires are most likely to occur during the dry season, which extends from July through September in this particular region. During this season, there is a prevailing westerly wind throughout the day. There are frequent lightning bursts that cause wildfires. The Forest Service wants to deploy four fire-fighting units to control fires during the next dry season. Each unit consists of 10 firefighters, one pickup truck, one dump truck, one water truck (50,000 liters), and one bulldozer (w/ truck and trailer). The unit has chainsaws, hand tools, and other fire-fighting equipment. The people can be quickly moved by helicopter within the wilderness area, but all the equipment must be driven via the existing firebreaks. One helicopter is on standby at all times throughout the dry season. Your task is to determine the best distribution of fire-fighting units within the wilderness area. The Forest Service is able to set up base camps for those units at sites anywhere within the area. In addition, you are asked to prepare a damage assessment forecast. This forecast will be used to estimate the amount of wilderness likely to be burned by fire as well as acting as a mechanism for helping the Service determine when additional fire-fighting units need to be brought in from elsewhere.
Gas Prices, Inventory, National Disasters, and the Mighty Dollar
It appears from the economic reports that the world uses gasoline on a very short supply and demand scale. The impact of any storm, let alone Hurricane Katrina, affects the costs at the pumps too quickly. Let’s restrict our study to the continental United States. Over the past six years, Canada has been the leading foreign supplier of oil to the United States, including both crude and refined oil products. (Petroleum Supply Monthly, Table S3 - Crude Oil and Petroleum Product Imports, 1988-Present. See page 5 for Canadian exports to the United States.)
Canada was the largest foreign supplier of oil to the United States again in 2004, for the sixth year running (from1999, when the country displaced Venezuela, to 2004 inclusive).
In 2002, Canada supplied the United States with 17 percent of its crude and refined oil imports — more than any other foreign supplier at over 1.9 million barrels per day.
Western Canadian crude oil is imported principally by the U.S. Midwest and the Rocky Mountain states.
Eastern Canada's offshore oil is imported principally by the U.S. East Coast states, and even by some Gulf Coast states.
Many refiners are buying enough to serve motorists' current needs, but not enough to rebuild stocks. "They are looking to buy the oil when they need it,” according to The Washington Post. "When they are uncertain about the future, they hold back." (The Washington Post: Crude Oil Imports to U.S. Slow With War 3/31/03.) Build a better model for the oil industry for its use and consumption in the United States that is fair to both the business and the consumer. You can build your model based on a peak day. Write a letter to the President’s energy advisor summarizing your findin
How fair are major league baseball parks to the players?
Consider the following major league baseball parks: Atlanta Braves, Colorado Rockies, New York Yankees, California Angles, Minnesota Twins, and Florida Marlins. Each field is in a different location and has different dimensions. Are all these parks “fair”? Determine how fair or unfair is each park. Determine the optimal baseball “setting” for major league baseball.
Outfield Dimensions Franchi Left se Angels Braves Rockie s Yankee s Twins 318 343 399 385 385 408 408 404 385 367 385 314 327 345 8 13 8 347 390 415 375 350 8 Left Center Field 408 401 Right Center 361 390 Right Field 330 330 Left Field 8 8
Wall Height Center Fi eld 8 8 Right Field 18 8 Area of Fair Ter 110,000 115,000
Field Center 330 335 376 380
7 13 8
10 23 8
113,000 111,000 115,000
Modeling Ocean Bottom Topography
A marine survey ship maps ocean depth by using sonar to reflect a sound pulse off the ocean floor. Figure A shows the ship’s location at B on the surface of the ocean. The sonar apparatus aboard the ship is capable of emitting sound pulses in an arc measuring from 2 to 30 degrees. In two dimensions this arc is shown within Figure A triangle by the dashed lines and the solid lines BA and BC. , and the emanating sound pulses are displayed by
When a sonar sound pulse hits the bottom of the ocean, the pulse is reflected off the ocean bottom the same way a billiard ball is reflected off a pool table; that is, the angle of incidence reflection equals the angle of
as illustrated in Figure B. Since the ship is moving when the sound pulse is emitted, it will
pick up a reflected sound pulse at location F in this picture. The actual depth of the water is the length of BD in Figure A.
Oceanography vessels usually travel at a speed of 2m/s while Navy vessels travel at 20m/s. The sonar apparatus aboard these ships is capable of emitting sound pulses in an arc measuring from 2 to 30 degrees. The typical speed at which a sonar sound pulse is emitted is 1500m/s. Devise a model for mapping the topography of the ocean bottom. Write a letter to the science editor of your local paper summarizing your findings.
Motel Cleaning Problem
Motels and hotels hire people to clean the rooms after each evening’s use. Develop a mathematical model for the cleaning schedule and use of cleaning resources. Your model should include consideration of such things as stay-overs, costs, number of rooms, number of rooms per floor, etc. Draft a letter to the manger of a major motel or hotel complex that recommends your model to help them in the management of their operation.
Consider a school where most of the students are from rural areas so they must be bused. The buses might pick up all the students and go to the elementary school and then continue from that school to pick up more students for the high school. A clear alternative would be to have separate buses for each school even though they would need to trace over the same routes. There are, of course, restrictions on time (no student should be in the bus more than an hour), drivers, equipment, money and so forth. How can you set up school bus routes to optimize budget dollars while balancing the time on the bus for various school groups? Build a mathematical model that could be used by various rural and perhaps urban school districts. How would you test the model prior to implementation? Prepare a short article to the school board explaining your model, its assumptions, and its results.
Skyscrapers vary in height , size (square footage), occupancy rates, and usage. They adorn the skyline of our major cities. But as we have seen several times in history, the height of the building might preclude escape during a catastrophe either human or natural (earthquake, tornado, hurricane, etc). Let's consider the following scenario. A building (a skyscraper) needs to be evacuated. Power has been lost so the elevator banks are inoperative except for use by firefighters and rescue personnel with special keys. Build a mathematical model to clear the building within X minutes. Use this mathematical model to state the height of the building, maximum occupation, and type of evacuation methods used. Solve your model for X = 15 minutes, 30 minutes, and 60 minutes.
The Art Gallery Security System
An art gallery is holding a special exhibition of small watercolors. The exhibition will be held in a rectangular room that is 22 meters long and 20 meters wide with entrance and exit doors each 2 meters wide as shown below. Two security cameras are fixed in corners of the room, with the resulting video being watched by an attendant from a remote control room. The security cameras give at any instant a “scan beam” of 30°. They rotate backwards and forwards over the field of vision, taking 20 seconds to complete one cycle. For the exhibition, 50 watercolors are to be shown. Each painting occupies approximately 1 meter of wall space, and must be separated from adjacent paintings by 1 meter of empty wall space and hang 2 meters away from connecting walls. For security reasons, paintings must be at least 2 meters from the entrances. The gallery also needs to add additional interior wall space in the form of portable walls. The portable walls are available in 5-meter sections. Watercolors are to be placed on both sides of these walls.To ensure adequate room for both patrons who are walking through and those stopped to view, parallel walls must be at least 5 meters apart throughout the gallery. To facilitate viewing, adjoining walls should not intersect in an acute angle. The diagrams below illustrate the configurations of the gallery room for the last two exhibits. The present exhibitor has expressed some concern over the security of his exhibit and has asked the management to analyze the security system and rearrange the portable walls to optimize the security of the exhibit. Define a way to measure (quantify) the security of the exhibit for different wall configurations. Use this measure to determine which of the two previous exhibitions was the more secure. Finally, determine an optimum portable wall configuration for the watercolor exhibit based on your measure of security.
A ladder 5 meters long is leaning against a vertical wall with its foot on a rug on the floor. Initially, the foot of the ladder is 3 meters from the wall. The rug is pulled out, and the foot of the ladder moves away from the wall at a constant rate of 1 meter per second. Build a mathematical model or models for the motion of the ladder. Use your model (or models) to find the velocity at which the top of the ladder hits the floor and the distance the top of the ladder will be from the wall at the moment that it hits the ground.
Major thoroughfares in big cities are usually highly congested. Traffic lights are used to allow cars to cross the highway or to make turns onto side streets. During commuting hours, when the traffic is much heavier than on any cross street, it is desirable to keep traffic flowing as smoothly as possible. Consider a two-mile stretch of a major thoroughfare with cross streets every city block. Build a mathematical model that satisfies both the commuters on the thoroughfare as well as those on the cross streets trying to enter the thoroughfare as a function of the traffic lights. Assume there is a light at every intersection along your two-mile stretch. First, you may assume the city blocks are of constant length. You may then wish to generalize to blocks of variable length
What Is It Worth?
In 1945, Noah Sentz died in a car accident and his estate was handled by the local courts. The state law stated that 1/3 of all assets and property go to the wife and 2/3 of all assets go to the children. There were four children. Over the next four years, three of the four children sold their shares of the assets back to the mother for a sum of $1300 each. The original total assets were mainly 75.43 acres of land. This week, the fourth child has sued the estate for his rightful inheritance from the original probate ruling. The judge has ruled in favor of the fourth son and has determined that he is rightfully due monetary compensation. The judge has picked your group as the jury to determine the amount of compensation. Use the principles of mathematical modeling to build a model that enables you to determine the compensation. Additionally, prepare a short one-page summary letter to the court that explains your results. Assume the date is November 10, 2003.