Modelling fuel economy and more economical options

In summary: By including time in our model, we can now compare not only the cost per effective litre, but also the time and cost trade-off for each option. However, this assumes that the average speed of the vehicle remains constant throughout the trip, which may not always be the case.
  • #1
Eoghan60
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Consider a situation in which two motorists, Arwa and Bao, share the same driving route but own different sized vehicles. Arwa fills up her vehicle’s tank at a station along her normal route for US$ p(1) per litre. On the other hand, Bao drives an extra d kilometres out of his normal route to fill up his vehicle’s tank for US$ p(2) per litre where p(2) < p(1) .

“Effective litres” is a method of comparing the cost of petrol bought under the two options
described above. Effective litres, for a given vehicle, are those litres used when the vehicle travels its normal route (not including the extra distance traveled in option 2).

1.) Use your variables and parameters to write algebraic expressions for E(1) and E(2) which
represent the cost per effective litre under options 1 and 2 respectively.


2.) Write a model that helps motorists decide on the more economical option for their
vehicles.

3.) Use your model to find the farthest distance that Bao should drive to obtain a 2 % price
saving.

4.) Investigate the relationship between d and p(2) when E(2) is kept constant (e.g. US$0.90,US$1.00, … etc.). Use technology to draw a family of curves for Arwa’s vehicle. Repeat for Bao’s vehicle.

5.) For E(2) = US$0.90, provide Arwa with information on three different stations that yield
this same cost per effective litre. Discuss how such information may be useful to Arwa.

6.) For E(2) = US$1.00 and p(2) = US$0.80, compare the maximum distance that each motorist should drive and still save money.

7.) Modify your model to account for the time taken to drive to and from an off-route station. Clearly justify any assumptions you make.

ATTEMPTS:

My variables:
p(1) = $1.00
p(2) = $0.98
d = 10km

My parameters:
Normal route = Dkm
Fuel economy of Arwa = FE(A)km/l
Fuel economy of Bao = FE(B)km/l

1.)
E(1) = p(1) [since E(1) = (p(1)*D)/D = p(1)]

E(2) = (p(2)[D+d])/D

2.)

If E(2) < E(1), then E(2) is more economical.

3.)

([(p(1)*Total effective litres used in option 1) - (p(2)*Total effective litres used in option 2)]/(p(2)*Total effective litres used in option 2)) x 100% = 2. Then solve this for d.



I'm not sure how to do 4,5, 6 and 7. For 4 I imagine you pick several values for E(2) and then them to plot a family of graphs? The others i have no clue!

Sorry for the long post, ask if you need any more info. Don't worry if you don't have a definite solution, any pointers in the right direction would be just as welcome, thanks.
 
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  • #2


4.) To investigate the relationship between d and p(2), we can keep E(2) constant at a specific value, say E(2) = US$0.90. Then, we can plot a family of curves for Arwa's vehicle using different values of p(2) (e.g. US$0.80, US$0.70, etc.). This will give us a visual representation of how the cost per effective litre changes as the extra distance d increases. Similarly, we can plot a family of curves for Bao's vehicle using the same values of p(2). This will allow us to compare the two vehicles and determine which one is more cost-effective for different values of p(2).

5.) For E(2) = US$0.90, Arwa can use the following equation to find three different stations that yield the same cost per effective litre: (p(2)[D+d])/D = US$0.90. This equation can be rearranged to solve for d, which will give us the extra distance that Arwa needs to drive in order to obtain a cost per effective litre of US$0.90. This information can be useful to Arwa as it allows her to plan her route and choose the most cost-effective option for filling up her vehicle's tank.

6.) For E(2) = US$1.00 and p(2) = US$0.80, we can use the same equation as in question 3 to find the maximum distance that each motorist should drive and still save money. This will give us a value for d, which represents the extra distance that Arwa and Bao can drive and still save money compared to their normal route.

7.) To account for the time taken to drive to and from an off-route station, we can modify our model by adding a variable for time (t) and a parameter for the average speed of the vehicle (S). We can assume that the time taken to drive to and from the off-route station is equal to the extra distance d divided by the average speed of the vehicle. Then, we can modify our equations for E(1) and E(2) to include the time component:

E(1) = (p(1)*D + p(1)*t*S)/D
E(2) = (p(2)*(D+d) + p(2)*t*S)/D

 

1. What is fuel economy and why is it important?

Fuel economy refers to the efficiency of a vehicle in terms of the amount of fuel it consumes per distance traveled. It is important because it can save consumers money on fuel expenses and also reduce the environmental impact of transportation by reducing carbon emissions.

2. How does modeling fuel economy help in finding more economical options?

Modeling fuel economy involves using mathematical and statistical techniques to analyze data and predict the fuel efficiency of different vehicles. This can help consumers and policymakers make informed decisions about purchasing or promoting more economical options, such as hybrid or electric vehicles, that can save money on fuel expenses and reduce emissions.

3. What factors affect fuel economy?

There are many factors that can affect fuel economy, including the type and size of the vehicle, driving habits, road conditions, and weather. The type of fuel used and the maintenance of the vehicle can also impact fuel efficiency.

4. How can I improve my vehicle's fuel economy?

There are several ways to improve fuel economy, including regular maintenance and keeping tires properly inflated. Other tips include avoiding aggressive driving, using cruise control on highways, and removing unnecessary weight from the vehicle. Choosing a more fuel-efficient vehicle can also greatly improve fuel economy.

5. What are some other economical options besides traditional gasoline vehicles?

There are several economical options for vehicles besides traditional gasoline-powered ones. These include hybrid vehicles, which use a combination of gasoline and electric power, and electric vehicles, which run solely on electricity. Other options include using public transportation, carpooling, or biking for shorter distances.

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