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.

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.

Thousands of species of plants and animals face threats that could result in their extinction, while biodiversity conservation actions that could save them are often available. Conservation managers face difficult decisions when there is limited funding for biodiversity conservation. Managers need to decide which projects they should fund to best achieve their objectives, not the least of which is to save the most species. This is particularly important if the benefits of certain conservation actions vary across projects, the costs of these actions for specific projects differ, and the available funding for these actions is considerably less than is needed to support all proposed projects.

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.

The United States can address its national carbon footprint in two ways: by reducing carbon dioxide emissions or by increasing carbon dioxide consumption (sequestration). Assume that the total U.S. carbon dioxide emissions are capped at 2007-2008 levels indefinitely. What should the U.S. do to increase carbon dioxide consumption to achieve national carbon neutrality with minimal economic and cultural impact? Is it even possible to achieve neutrality? Model your solution to show feasibility, effectiveness, and costs. Prepare a short summary paper for the U.S. Congress to persuade them to adopt your plan.

During events at your school, students operate a concession stand to raise money in support of student activities. Your team is in charge of the concession stand for this year. Your team will sell food and other items at the stand. There are 10 monthly events this year.

Gas prices fluctuate significantly from week to week. Consumers would like to know whether to fill up the tank (gas price is likely to go up in the coming week) or buy a half tank (gas price is likely to go down in the coming week).

Consider the following cases:

Consumer drives 100 miles per week
Consumer drives 200 miles per week
Assume:

Gas tank holds 16 gallons and average mileage is 25 miles/gallon => 400 miles/tank
Consumer buys gas once a week

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.

A parachute is made from thin, lightweight fabric, support tapes and suspension lines. The lines are usually gathered through cloth loops or metal connector links at the ends of several strong straps called risers. The risers in turn are attached to the harness containing the load.

Organizing international meetings is not easy in many ways, including the problem that some of the participants may experience the effects of jet lag after recent travel from their home country to the meeting location which may be in a different time-zone, or in a different climate and time of year, and so on. All these things may dramatically affect the productivity of the meeting.

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 by , and the emanating sound pulses are displayed by the dashed lines and the solid lines BA and BC.

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.

A great deal of preparation must take place before a movie can be filmed. Important sets and scenes need to be identified, resource needs must be calculated, and schedules must be arranged. The issue of the schedule is the focus of the modeling activities. A large studio has contacted your firm, and they wish to have a model to allow for scheduling a movie. You are asked to answer the questions below. You should provide examples and test cases to convince the movie executives that your model is effective and robust.

Mathematical modeling involves two equally important steps - building models based on real world situations and interpreting predictions made by those models back in the real world. This problem places equal emphasis on both steps.

We are at the start of the 2008 U.S. presidential elections, and one important area of debate is sure to be the national debt. As high school students, you have a particular interest in this subject since you are the people who will pay off or at least manage the national debt in the future. The rate at which the national debt changes depends on the difference between federal income (primarily taxes) and federal expenditures. Your first task is to build a model that can be used to help understand the national debt and make forecasts based on different assumptions. As usual, modeling involves a balance between so much complexity that the model may be intractable and so little complexity that it is unrealistic and useless. Your model needs, at the very least, to allow you to consider different tax policies and different expenditure policies.

Honeybees, along with a few other key animals, are critical to human existence on our planet. Along with honey production, these insects provide the vital role of pollination of many trees and plants that provide food for our survival. In 2007, the term Colony Collapse Disorder (CCD) was created to describe the decline of honeybee populations around the world.[1] Bee decline can be attributed to factors such as viruses, pesticides, predators, habitat destruction, and environmental conditions.

In 2014, the world saw the infectious Ebola virus spreading in western Africa. Throughout human history, epidemics have come and gone with some infecting and/or killing thousands and lasting for years and others taking less of a human toll. Some believe these events are just nature’s way of controlling the growth of a species while others think they could be a conspiracy or deliberate act to cause harm. This problem will most likely come down to how to expend (or not expend) scarce resources (doctors, containment facilities, money, research, serums, etc...) to deal with a crisis.

A group of four friends want to go to Polygon Paradise Park next summer. Polygon Paradise Park is a small amusement park open from 9am to 9pm. The park has ten rides of different types and thrill levels. The Trapezoid Show, a location with several different performances throughout the day, offers a circus acrobatics show, a magic show, and evening fireworks. The Games building has both arcade and carnival games. Food options include the Triangle Restaurant, the hot dog stand, and the ice cream cart. The hot dog stand and the ice cream cart are “take away” options, while the Triangle Restaurant requires 30-60 minutes to sit at a table and enjoy a meal. The restrooms (WC) are centrally located in the park for easy access. Visitors can purchase souvenirs in the gift shop located near the park entrance.

The next Olympic Games are being hosted by Paris, France and will be held 26 July - 11 August 2024. The International Olympic Committee (IOC) wants to prepare by ordering enough medals and flags for the medal ceremonies. The IOC wants to make sure they have enough medals and flags for all the ceremonies to avoid being embarrassed. The IOC also does not want to waste money, however, by ordering too many medals and flags that will not be used. Therefore, the IOC has asked your team at MidMCM Olympic Consultants to provide models they can use to determine how many medals and flags they will need.

Have you ever been traveling down a highway when you see one or both of these signs?

In some cases, the road simply loses a lane as it enters a less traveled section. In other cases, the road narrows because of construction or roadwork. When this happens on a fairly busy highway, it may be a recipe for road rage*.

In athletics, one of the possible distances to run is 15,000 meters or 15k (in the picture you see the leader in an annual 15k - race in the Netherlands. Please see Wikipedia article below). For this type of run, 15k on a street track, there is a world record, as there are records for all other distances that are run in athletics (e.g. the marathon). In such a race, the organizing committee will usually pay a significant amount of money as a bonus to the winner if he or she succeeds in setting a new world record. These amounts of money can get quite large in order to attract top runners: in the race shown in the picture there was a 25,000 euro bonus if the winner succeeded in improving the 15k world record – which, by the way, he (un)fortunately did not achieve. Had he done so, there would have been a major financial problem for the organizing committee, since they had not purchased any insurance.