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MATH IA- Filling up the petrol tank ARWA and BAO

Extracts from this document...

Introduction

[Math Hl Portfolio type 2 Samkit shah]

Filling up the Petrol Tank

In this world money is one of the most important things, and hence we try to save as much money as possible. Even drivers try to save as much money as possible. Some places sell fuel for lower prices than others. Is it more economical to travel a larger distance to get cheaper fuel or is it more economic to travel a shorter distance but to buy relatively expensive fuel? Let’s investigate!

There are two people Arwa and Bao, who share the same driving route r of 20km.

Let’s assume:

  • The two are neighbors and work at the same place.
  • They drive to work every day.
  • They buy fuel in the morning, before going to work, every second Monday, as they are busy people.
  • They used their car only to go to work and they work only from Monday to Friday.
  • They are cautious people and so they always have at least 5litres petrol as reserve. If it goes below that they will go and refuel at that instant.
  • They have enough petrol to reach the gas station initially. After reaching the gas station they buy exactly the fuel required to drive to work the next ten days and to drive to and from the station (if the gas station is not on the route).

Since Arwa and Bao work only Monday to Friday and use their car only to go to work, we can say that they fuel up every 10 working days (number of working days after which, they refuel is w working days).

...read more.

Middle

Note: The costs to Arwa and Bao per day and the money saved by Bao per day are rounded to two decimal places in the tables

MS Excel was used to make the tables. It was used as it allows us to see the data in an organized manner, and there are options called formulas, which allow us to calculate the values easily. Geogebra was used to make the graphs as the GDC does not have good resolutions and the scale is not present on the graph.

Seeing the effect of changing p1 on the money saved by Bao (with respect to Arwa), when p2 and d are kept constant at 0.98$ and 10km respectively

p1 (US$)

Money saved by Bao (with respect to Arwa)(US$)

Formula:(eq1)

1.00

1.64

1.10

3.86

1.20

6.09

1.30

8.31

1.40

10.53

image00.png

Note: The equation of the line is the equation of money saved by Bao(with respect to Arwa) .Therefore it is y=(400)x/(18)- (2×10+400)×0.98/(20)

From the graph we can see that as p1 increases, the money saved by Bao (with respect to Arwa) increases. The reason for this is that as p1 increases the cost of fuel to Arwa increases and so Bao saves more money when compared to Arwa. The graph is linear; therefore money save by Bao (with respect to Arwa) is directly proportional to p1.

Seeing the effect of changing p2 on the money saved by Bao (with respect to Arwa), when p1 and d are kept constant at 1.00$ and 10km respectively

p1 (US$)

Money saved by Bao (with respect to Arwa)(US$)

Formula:(eq1)

0.90

3.32

0.92

2.90

0.94

2.48

0.96

2.06

0.98

1.64

image01.png

Note: The equation of the line is the equation of money saved by Bao(with respect to Arwa) .Therefore it is y=(400)×1/(18)- (2×10+400)x/(20)

...read more.

Conclusion

Now Let’s find the maximum distance for Bao

∴∴2r×1.00$/20km<1.00$

r<10km

 Bao can no longer save money by just buying fuel from his normal route after the distance to work becomes equal to or greater than 10km.

Arwa is a really business and wonders whether saving money is worth the time. Let’s help him decide.

Arwa must have a money value for time as he is considering saving money or saving time.

Let this be s dollar/hour.

Money saved by taking detour(on an average per working day)

=2rp1/(f l)- 2p2(r +d× day/w)/f l

Extra time taken per working day due to detour

=(2×distance of detour/working day)/speed

=(2d day/w)/(40km/hour)        (Let’s assume the time taken per our in city is

constant at 40km/hour)

Therefore in money value time taken due to detour per day costs

= Extra time taken per working day due to detour × Arwa’s money value of time

=(2d day/w)/(40km/hour) ×s

Real benefit due to detour in money value

= Money saved by taking detour- in money value time taken due to detour per day costs

=2rp1/(f l)- 2p2(r +/w)/f l-(2d day/w)/(40km/hour) ×s

If the above equation is positive then Arwa should take the detour and if it is negative he shouldn’t.

Let’s consider the limitations of this model. This model also has the problems related to the assumption of refueling regularly. On top of that this model assume the speed in the city is constant at 40km/hour, which is not true. But there are so many uncontrolled variables, that a model cannot account for all of these variables. Even though these assumptions affect the accuracy of money saved, it can be used to get a fair idea of the benefits; hence I believe it is useful.

...read more.

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