Quick Notes: Project 1 Gas Station Problem - Work Together

Quick Notes Page 1project 1 Gas Station Problemwork Together In Your

Pick one of your routes to school to work with. Discuss with your group members other considerations that will go into your decision-making process about going off of your route to get gas. Let’s assume you drive to school every day. Map out your route and look at the gas stations that are on the way and how much the price of gas is at those stations.

Then look at other gas stations in the area that are priced lower than the stations along your route. Would it be more economical for you to drive the extra distance for the less expensive gas than to purchase gas along your route? Would it be worthwhile for you to drive to the gas station that is priced lower than the others? Develop a mathematical equation that will determine your cost to get gas along your route and the cost to drive further away for cheaper gas to make your decision. Give your reasoning for those answers.

• Let p represent the price per gallon at the station along our route and P the price per gallon at the station we are considering.

• Let D represent the distance in miles from the normal route that must be driven to get to the gas station (so a round trip is 2D miles).

• Let M represent the miles per gallon of the vehicle.

• Let T represent the number of gallons of gas we purchase when we buy gas (in our model, four gallons less than the tank size).

Pick a route to school for one of the people in your group to use for this project. If everyone in your group lives on campus or out of the country, pick a location in Brevard County to use as “home” and map your route from that location. Pick one of your vehicles to use for this problem. If no one in the group owns a vehicle, choose a vehicle you would like to “use to drive to school.”

Questions to Address

1. List all variables and their values.
2. Describe the mathematical model developed to find a solution.
3. What assumptions can you make about your model?
4. Which option did you determine is more economical? Why?
5. How do you know you have a good model?
6. Pick two different vehicles: one that gets more miles per gallon than your vehicle and one that gets less. Which option is more economical with these vehicles? Why? Show all work.
7. What do the results from above tell you about the sensitivity of your model?
8. What other considerations would contribute to your decision about which gas station to use? (Give your reasoning – whether it would be worthwhile or not to drive to the gas station that is priced lower than the others?)

Reflection

1. Explain the process your team used to develop a solution.
2. Explain the mathematics used to develop your team’s solution.
3. How did each of your teammates participate in the modeling process?
4. What did you learn from the other members of your team?
5. Did you revise your model at any point during the activity? If so, why? How did you fix the model?
6. Can you identify a math idea that was key to your ability to develop a model?
7. What advice would you give to a classmate (or yourself) prior to developing a mathematical model?
8. What was the most unexpected aspect of this project?

Paper For Above instruction

The Gas Station Problem presents an engaging mathematical approach to real-world decision-making involving economics and logistics. This project invites students to develop a model that helps determine whether it is more cost-effective to stick to their regular fueling route or to drive extra miles to purchase cheaper gas. The goal is to formulate a mathematical equation considering variables like fuel prices, distances, vehicle efficiency, and consumption, then analyze which option minimizes costs.

To begin, comprehensively listing all variables involved is essential. The key variables include: p, the price per gallon at the station along one's route; P, the lesser price at a different station located away from the regular route; D, the additional miles driven to reach the alternative station; M, vehicle fuel efficiency in miles per gallon; T, the number of gallons purchased, which is less than the tank capacity. The values assigned to these variables depend on actual data collected from mapping routes and gas stations. For instance, if the route includes stations charging $3.50 per gallon, and the alternative station offers gas at $3.00 per gallon, these values feed directly into the model.

The core of the mathematical model involves calculating the total cost of fueling along the normal route versus detouring for cheaper gas. The cost of fueling along the route is computed as the product of the price per gallon and the number of gallons, i.e., pT. For detouring, the total mileage extends by 2D miles (round trip), and the additional fuel cost considers the extra miles driven and the lower price P thus, total detour cost equals P T plus the additional miles cost, which is (2D / M) * P. The decision model compares these two total costs to determine the more economical choice.

Assumptions in this model include constant vehicle fuel efficiency regardless of driving conditions, uniform gas prices at each station, and the driver's willingness to split their fuel purchase. It assumes the additional miles driven do not significantly affect fuel economy, and the driver considers only monetary cost without factoring in time or convenience.

In analyzing the model, the option with the lower total cost is deemed more economical. For example, if the savings per gallon offset the extra miles driven, then detouring is justified. Conversely, if the cost of additional miles outweighs fuel savings, sticking to the regular route is preferable. Through sensitivity analysis, varying the price P or D illustrates how changes influence the decision, demonstrating the model's responsiveness to real-world fluctuations.

Introducing different vehicle efficiencies provides further insight. For a vehicle with higher miles per gallon, the additional mileage cost becomes less significant, making detours more attractive. Conversely, for less fuel-efficient vehicles, the added miles significantly increase costs, discouraging detours. Calculation examples with distinct vehicles confirm these trends, highlighting the importance of vehicle efficiency in cost analysis.

Additional considerations influencing the decision include time constraints, availability of fuel stations, and personal preferences for convenience or brand loyalty. Even if a gas station offers a lower price, the inconvenience of detouring might outweigh cost savings, especially for time-sensitive commuters.

Reflecting on the process, developing this model involved teamwork and collaborative problem-solving. Each member contributed by mapping routes, gathering data, and performing calculations. Using algebraic expressions and inequalities, the team formulated a cost comparison model, applying assumptions to simplify complexities. Iterative revisions enhanced the model's accuracy and relevance, such as adjusting for different vehicle efficiencies or more realistic fuel consumption scenarios.

A key mathematical idea in this project is the use of linear equations and inequalities to balance costs and distances, facilitating decision-making. Prior to modeling, advisable strategies include clearly defining variables, gathering accurate data, and testing the model's sensitivity to key changes. The most unexpected aspect was realizing how small fuel price differences can be overshadowed by extra driving costs, emphasizing the importance of comprehensive analysis instead of relying solely on fuel prices.

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