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# Mathematics IA - Particles

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Introduction

In this investigation I will be studying the case of an infection of particles. I will be looking into, and analyzing, how the particles work when they first enter the body, what effect the response of the immune system has, how medication is delivered and maintained, as well as death and recovery. Furthermore, I will be altering my investigation models to cater to a young child as opposed to an adult.

First I will look at the initial phase of the infection – the part where the particles enter the body and replicate yet none are expelled because the immune system hasn’t responded. To determine how long it will take before the immune system responds I need to create a basic formula:

a(rn)

In this formula, a represents the initial amount of particles and r represents the ratio at which they multiply every 4 hours – just like they are used in sequences. n represents how many times they multiply, which is once every 4 hours.

Considering the case of a young adult male, I presume that he is initially infected with 10,000 particles and that they double every hour. I also presume that the immune system responds when the particle count reaches 1,000,000. Therefore in order

Middle

÷ 60 = 0.0416)% of the medicine is eliminated every minute. This enables to create the following equation:

D(0.9996240) + D(0.9996239) + … + D(0.99961) + D

This formula helps determine the dosage because D represents the dosage per minute, and multiplying with 0.999583 is the equivalent of removing 0.0416%. This number has a power of 240 because 0.0416% is removed 240 times by the end. For the next dose injected a minute later, this takes place 239 times because it is in the body for 1 minute less. Similarly, the dose injected 1 minute before the end of the 4 hours only has a power of 1 because 0.0416% is only removed one time before the end.

This equation can then be rearranged to put the smallest powers at the beginning and the largest powers at the end but either way I have a geometric sequence. I then use the formula to find the sum of a geometric sequence:

a(rn – 1) ÷ (r – 1) where a is the initial term (D, if the equation is rearranged) and r is the ratio of multiplication (0.9996, if the equation is rearranged) and n is the number of times the ratio is multiplied (240).

This equation must multiply out to 90 so that 90 micrograms are ensured in the system at the end of the four hours. Plugging the values in I get the following:

D(0.9996240 – 1) ÷ (0.9996 – 1) = 90

D(-0.095) ÷ (-0.000416) = 90

D(-0.095) = 90 × (-0.000416) = -0.038

D = (=0.038) ÷ (-0.095) = 0.39 micrograms

Conclusion

I will now explain how I derived the general formula. First I presume that I are just finding a general formula for the time period after the immune system kicks in so I get the following:

{[1,000,000(1.60.25) – 50,000](1.60.25) – 50,000}(1.60.25) – 50,000 …

Opening this up I get the following:

1,000,000(1.60.25)(1.60.25)(1.60.25)–50,000(1.60.25)(1.60.25)–50,000(1.60.25)–50,000 …

For the first part I can immediately make out that the number of (1.60.25) attached to the 1,000,000 are equal to the number of hours passed making it, where n is the number of hours passed:

1,000,000(1.60.25n)

For the second half we can factorize 50,000 out leaving us with:

X – 50,000{(1.60.25)(1.60.25) + (1.60.25) + 1}

X – 50,000{(1.60.25(2)) + (1.60.25(1)) + (1.60.25(0))}

I then see a geometric equation forming with a being 1 because (1.60.25(0)) = 1 and r being (1.60.25). Using the sum of a geometric formula:

a(rn – 1) ÷ (r – 1)

1[(1.60.25n) – 1] ÷ [(1.60.25) – 1]

From this we can derive the general formula after the immune system kicks in to be:

1,000,000(1.60.25n) – 50,000[(1.60.25n) – 1] ÷ [(1.60.25) – 1])

We  can then use this format to derive a general formula that would work for any numbers:

P(r0.25n) – e[(r0.25n – 1) ÷ (r – 1)]

In this formula P is the starting number of particles (so 10,000 if we start from when he was first affected, or 1,000,000 if we start from when the immune system kicks in). r is the rate of multiplication every 4 hours, and e is the number of particles eliminated from the body every hour. From this formula we can plug in any values for any of the factors and determine the number of particles.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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