Maths Modelling (Applied Maths Optimisation- how to make into terms of x and y) -

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alex.kundert
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Hello all and many thanks for your help with the maths problem
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alex.kundert
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Attachment 720938

Please may you help me with part B and C of this question? Specifically just how to model the problem in terms of x and y and or equations


Thx all
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RDKGames
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(Original post by alex.kundert)

Please may you help me with part B and C of this question? Specifically just how to model the problem in terms of x and y and or equations


Thx all
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alex.kundert
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(Original post by RDKGames)
Impossible to read. Post a clearer pic.
Hiya

Please find a copy of the question below - I have copied it out in words



A company that runs safaris to see the Northern Lights plans to construct luxury igloo accommodation on a site somewhere on a straight line between two settlements, Kilpi and Lappi. Lappi is 12km due east of Kilpi. The company estimates that it will take tourists three times as often to Kilpi as it will to Lappi. It wishes to position the accommodation such that the amount of travel from the igloos to the settlements is minimized.


(b) Create a mathematical model for the company’s problem. State any assumptions that you make, and define any variables that you use. Solve the problem, and interpret your solution.


(c) Before it began construction of the site, the company carried out a survey of tourists and realized that there was a demand to go to a nearby reindeer farm. The farm is 3km east and 6km north of Lappi. The company estimated that it would need to make the same number of journeys to the farm as it did to Kilpi. Revise your model from part (b) to take account of the company’s situation, and no longer restricting the site of the igloos to being on the straight line between Kilpi and Lappi. State any additional assumptions that you make. Determine the position where the company should site the igloos so that the total travelling distance to the settlements and the farm is minimized.
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MindTheGaps
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I'll help you with the model formula for B because you should be able to work then from there. You can get a number for time travelled, T, in terms of distance from Kilpi, x. The three is to account for the frequency of visits.

T = 3x + (12-x)

Then differentiate to find the x which minimises the time.

Technically depending what quantity you were actually representing (average time travelled per round trip, etc.) you might come up with a number which are a scalar multiple of the formula above, but this doesn't effect the answer here.

You will notice (and I suspect this is what is confusing you) that  \frac{dT}{dx} = 2 which is a (positive) scalar! This tells you that there is no minimum, the function just keeps decreasing as you decrease x towrds zero. Hence, the best way to situate the site is actually at Kilpi.

For C, try coming up with an equivalent formula in terms of two coordinates, x and y. Yes you will need to use Pythagoras to get the time to each site. Then differentiate by x and y to get simultaneous equations to solve. I don't think you'll get so trivial an answer here.
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alex.kundert
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(Original post by Rinsed)
I'll help you with the model formula for B because you should be able to work then from there. You can get a number for time travelled, T, in terms of distance from Kilpi, x. The three is to account for the frequency of visits.

T = 3x + (12-x)

Then differentiate to find the x which minimises the time.

Technically depending what quantity you were actually representing (average time travelled per round trip, etc.) you might come up with a number which are a scalar multiple of the formula above, but this doesn't effect the answer here.

You will notice (and I suspect this is what is confusing you) that  \frac{dT}{dx} = 2 which is a (positive) scalar! This tells you that there is no minimum, the function just keeps decreasing as you decrease x towrds zero. Hence, the best way to situate the site is actually at Kilpi.

For C, try coming up with an equivalent formula in terms of two coordinates, x and y. Yes you will need to use Pythagoras to get the time to each site. Then differentiate by x and y to get simultaneous equations to solve. I don't think you'll get so trivial an answer here.

Thank you so much!!
That is very greatly appreciated and has helped me with my understanding of part B. I am still struggling with C - would it please be okay if you could help me with the model in terms of a mathematical formulae for part C too?

Cheers and thanks again,
Alex
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alex.kundert
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Can anybody please help me with part c ? Thank you very much
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alex.kundert
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Hi there

Please may somebody help me with the following question? I have trouble applying maths and ideally I just need someone to help me model the problem in terms of x and y so to speak so I can then progress with the detail of the question ( such as differentiating it). Any help would be greatly appreciated, thank you. The question is as follows:

A company that runs safaris to see the Northern Lights plans to construct luxury igloo accommodation on a site somewhere on a straight line between two settlements, Kilpi and Lappi. Lappi is 12km due east of Kilpi. The company estimates that it will take tourists three times as often to Kilpi as it will to Lappi. It wishes to position the accommodation such that the amount of travel from the igloos to the settlements is minimised.

(c) Before it began construction of the site, the company carried out a survey of tourists and realized that there was a demand to go to a nearby reindeer farm. The farm is 3km east and 6km north of Lappi. The company estimated that it would need to make the same number of journeys to the farm as it did to Kilpi. Revise your model to take account of the company’s situation, and no longer restricting the site of the igloos to being on the straight line between Kilpi and Lappi. Determine the position where the company should site the igloos so that the total travelling distance to the settlements and the farm is minimized.

You may assume that if your solution process requires the minimization of a non-linear function of more than one variable, then the function has a single local minimizer, which occurs at a stationary point in the region under consideration.
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alex.kundert
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Plz help ?
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