# Exploration in Maths: Comparing Aircraft Dimensions for Optimal Flight Profile

• shwareb
In summary, the student was interested in modeling airplane flights and optimizing them for the least amount of fuel consumed, but was unable to complete the project due to difficulty understanding the calculus involved. He eventually decided to focus on comparing different dimensions of aircrafts instead.
shwareb
* I know this is a very long post but it would mean the world to me if you could read it. I would really appreciate it.

Wll hi guys, so I'm in the final year of my IB diploma programme and in maths we have a component known as internal assessment in which you're supposed to write a so called Mathematical Exploration:
"This mathematical exploration is a short report written by the student based on a topic chosen by him or her, and it should focus on the mathematics of that particular area.The final report should be approximately 6 to 12 pages long. The report should include a detailed bibliography, and sources." - IB

Yeah so I initially started to think about ideas back in July, and already then I knew I wanted to do something calculus orientated. Some people suggested that I could do "mixing problems" and set up a differential equation to describe the rate of change in a contaminant, or even a lake because a real world example is what I should really go for.

I should use calculus to explore real life problems. So I actually found a document of a person who had set up their own model describing how the rate of contaminant changed over time using Calculus and then he applied this to some lakes in real life which I thought was cool.
So I showed my teacher this document and she said I couldn't do that as I would take somebody else model and it wouldn't be my authentic work. So I had to switch idea.
Now for the past months together with my struggle in the IB I've gotten many ideas, but I've always been centered around calculus.

The teacher sat down with me and told me what I was interested in and we dug deap until we decided to do something with airplanes.
Airplanes are a huge fascination of mine. I love the engineering, the structure, it's greatness. I even love airports. The strange thing is that I'm super afraid of flying airplanes (partly because I've watched all Mayday episodes).

So initially I thought let's create a model to optimize a flight profile for the least amount of fuel consumed. I then thought I would look at which airplanes to take for different trips. At which altitude you should fly, what speed you should have, when should you start descent etc... to optimize flight profile for least amount of fuel consumed.
This turned out to be really hard as there were many variables involved and I had to give up that idea after to 15+ hours research if not more...

Then I went on by thinking well let's look at airplane descent and try to model the descent of an airplane. I would then look at different airplanes at different airports.
Which speed should the airplane have initially when should it start to descent etc...
I found a model on the internet which did model this but after 8 hours with it I gave up, I thought it was a bit too hard to comprehend.
I thought I could use the model and then compare with real data and probably see that they were not the same and then discuss this.
In theory it was a great idea but once again I couldn't pull through.

So then I took a step back and thought what made me interested in planes, and then I thought about the dimensions, and structure etc...
The wing loading and flexibility of airplanes are super important to me when I choose a trip by airplane. Since an increase in wing loading and flexibility decreases any turbulence felt by me in the fuselage and that is a dealbreaker for me.
I also always sit by the wing as it's statistically the safest place to sit at (but also the noisiest).
In the airplane the three structures which are the most important to me and the ones I worry about the most, are the wings, the engines, and the rudder/stabilizer.
I'm especially scared that the rudder/stabilizer will malfunction or get stuck, as it will often have fatal consequences.

Now I said this to my teacher and she was like: "You should compare the dimensions of different aircrafts, perhaps those big ones with small ones, and surely you can get some calculus there"
That was what she said, now after spending about 25 hours trying to find out what to do I'm getting really demoralized and at the same time the school is giving me a rough time (understandably so but really I have been trying a lot)
I have collected dimensions of four different aircrafts, two long-haul and two short-haul. At least I've tried to find as much data as I can. I especially got a lot for the Airbus A380 but now I'm like what am I supposed to do with the data, which essentially is only dimension data. Where can I use my calculus, and then what...

After spending more than 60 hours on this I'm getting really demoralized and sad really, and I can feel the pressure increasing. What should I do guys really feeling hopeless at the moment. The school is really getting sick of me too.

In this "Maths Exploration" the maths definitely does not have to be complicated, this is a very good example of a maths exploration which scored a perfect 20/20

The assessment criteria are these and I would really appreciate it if you took a look at them.

I really need your help guys :(

I just need my topic and then the writing stage will go much faster

Cheers guys!

P.S. My maths knowledge is equivalent to that of a Maths HL IB student although I only study Maths SL.

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Calculating optimal airplane descents/ascents and such is a little complicated; probably too much so for an introductory calculus student (at least as far as I am aware).

Have you done multivariable calculus, or only single variable calculus? You could try to model flight planning with a primitive model. As an extremely simple example:

A plane flies back and forth between airport A and airport B. For each flight the airline makes 500,000 dollars in revenue. The amount of fuel required to fly from one airport to the other is proportional to the square of the velocity of the plane (probably, at least this is true given the typical simple model for aerodynamic drag). How fast should the plane fly in order to maximize profit? If you fly too fast you'll spend too much money on gas, and too slow and you won't get as many flights in as you want - the solution to the problem can be found using calculus

Then you can add wrinkles, like what if when you fly more flights you get less money per flight (which is probably true), and what if also you take into account that you use less fuel when you are carrying less additional weight> What if there is some wind going from airport A to airport B continuously (which will be true for any pair of airports located east/west of each other) - what speed should you fly in each direction?

At some point the equations will become too hard to solve algebraically, which is probably when you stop adding wrinkles to the problem.

Sorry to hear it's so discouraging. I think what you're going through seems pretty typical for doing research. I think it's not unheard of to spend more time narrowing in on a topic than it is doing the actual research.

Are you still looking for a new topic or are you pretty set on the planes thing?

As far as the planes goes... (I could be wrong, I tend to do more pure maths)...The only thing that comes to mind for me is that things on that scale don't tend to scale up exactly proportionally. i.e. if I want to build one plane twice as big as another, I can't just double everything. There are other factors to consider.

Sometimes books on Diff EQ and Calculus have a "projects" section that can generate ideas. Or even just looking at some of the problems themselves - the ones near the end of the problem set or end of the chapter.

I'm not sure what I've said was very helpful here, but I wanted to at least let you know somebody was reading. :)

-Dave K

Office_Shredder said:
The amount of fuel required to fly from one airport to the other is proportional to the square of the velocity of the plane

Small correction: I think Office_Shredder meant "the rate of burning fuel (i.e. mass/second or volume/second) is proportional to ..."

If you fly slowly, you burn fuel slowly but the flight between two airports takes a long time. If you fly fast, the flight time is shorter but you burn fuel faster.

(And note this is a very simplified model, so it might not work too well. At very low flying speeds, you actually need more engine power to fly slower, which is the reverse of "common sense". But go to an air show, and listen to the engine noise from military jet fighter when it's flying straight and level but insanely slowly, with the nose pointing 45 degrees up...

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AlephZero said:
Small correction: I think Office_Shredder meant "the rate of burning fuel (i.e. mass/second or volume/second) is proportional to ..."

If you fly slowly, you burn fuel slowly but the flight between two airports takes a long time. If you fly fast, the flight time is shorter but you burn fuel faster.

The total energy that it takes to fly from A to B is equal to force*distance; the distance is constant and force is going to be proportional to the square of the velocity for air resistance. You'll actually burn fuel at a rate proportional to the cube of the velocity.

I'm not really bound by this topic, but I am scared to just go off in a new direction. The teachers are really making my life miserable.
Im in my final year of sixth form, I am in second year of the IB programme.

I have only done single variable calculus, so I sadly can't do anything about Multivariable Calculus?

shwareb said:
I have only done single variable calculus, so I sadly can't do anything about Multivariable Calculus?

I don't know. If you have a good grasp on single variable calculus, then multivariable calculus isn't that much harder. It's certainly doable if you're motivated enough.

shwareb said:
I'm not really bound by this topic, but I am scared to just go off in a new direction. The teachers are really making my life miserable.
Im in my final year of sixth form, I am in second year of the IB programme.

I have only done single variable calculus, so I sadly can't do anything about Multivariable Calculus?

The suggestions in my above post all required single variable calculus only, if they looked interesting/suitable as topics for you.

## 1. What is the purpose of exploring aircraft dimensions for optimal flight profile in mathematics?

The purpose of this exploration is to determine the most efficient and effective dimensions of an aircraft for achieving optimal flight performance. This can help improve fuel efficiency, reduce flight time, and enhance safety.

## 2. How does mathematics play a role in this exploration?

Mathematics is essential in this exploration as it involves data analysis, calculations, and modeling to determine the optimal dimensions. Mathematical concepts such as geometry, trigonometry, and calculus are used to analyze the relationships between different dimensions and their impact on flight performance.

## 3. What are some factors that are considered when comparing aircraft dimensions?

Some factors that are considered include the weight of the aircraft, its aerodynamic properties, the type and size of the engine, and the intended flight profile. These factors can affect the aircraft's lift, drag, and thrust, which ultimately determine its performance.

## 4. Can this exploration be applied to all types of aircraft?

Yes, this exploration can be applied to all types of aircraft, including commercial airplanes, military jets, and even drones. However, the specific dimensions and factors that are considered may vary depending on the type and purpose of the aircraft.

## 5. What are some potential benefits of using optimal aircraft dimensions?

Using optimal aircraft dimensions can lead to various benefits, including improved fuel efficiency, reduced emissions, increased speed and range, and enhanced safety. It can also potentially reduce operating costs and improve the overall performance and reliability of the aircraft.

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