• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Mathematics SL Portfolio Type II Modeling the amount of drug in a bloodstream

Extracts from this document...

Introduction

Mathematics SL

Portfolio

 Type II

Modeling the amount of drug in a bloodstream

                                                                                                                             November 30, 2007


The use of math in life is extremely beneficial; one can know or at least predict what to expect, why to expect it and how to find it. An example of an application of math in life would be medicine; doctors can calculate how much medicine to prescribe and how long it lasts or how long it takes for it to decay, therefore, how often should the patient take it.

In a case for treating Malaria, an initial does of 10 milligrams (μg) of a drug was given and an observation for a total period of 10 hours was done, the amount of drug was measured every half an hour, the results were plotted on the following graph(figure 1):

image02.png

image03.png

As can be seen, this graph shows the rate of the breakdown of the drug in the blood where the amount of drug in the bloodstream decreases with time; they are inversely proportional. In the following table are the numerical results of this observation taken from the previous graph:  

Time in hours

(t/hr)

Amount of drugs in μg

(Q/μg)

0.5

9.0

1.0

8.3

1.5

7.8

2.0

7.2

Time in hours

(t/hr)

Amount of drug in μg

(Q/μg)

2.5

6.7

3.0

6.0

3.5

5.3

4.0

5.0

4.5

4.6

5.0

4.4

5.5

4.0

6.0

3.7

6.5

3.0

7.0

2.8

7.5

2.5

8.0

2.5

8.5

2.1

9.0

1.9

9.5

1.7

10.0

1.5

...read more.

Middle

ec = b1

ec ≈ (0.829)1

c = ln(0.829)

c ≈ -0.2

Therefore, b ≈ e-0.2

And so, f(t) = 10.477e-0.2t where (t) is the time taken for the amount of drug to change in the bloodstream.

The function previously found will help in making a model to predict the change in the amount of drug in the bloodstream over long periods of time without having to expose the patients directly to the drug just in case something might go wrong. The following graph (figure 2) shows the difference between the function for the model and the data plotted in (figure 1) along with its curve-fit to raise the aware of the fact that a model does not 100% apply in real life.

image08.pngimage06.pngimage07.pngimage00.pngimage01.png

image09.png

As shown in (figure 2), the function is accurate enough to be used in making predictions as the curve is close enough to the plotted points and it also crosses the curve-fit at some points as well.

...read more.

Conclusion

image12.pngimage13.png

It can be seen that the drug will completely breakdown after less than 3 days. However, according to the model in (figure 5) below, if the patient continued to take the drug every six hours for a week, the amount of drug will always reach the same maximum amount of 15.0μg and a minimum amount of 4.5μg after a period of 60 hours, this concludes that the rate of decrease of the drug is almost proportional to the amount remaining.

image04.png

image05.png

Using a math application in such a case is greatly useful and helpful in order not to make errors in real life for if a patient, such as in the previous case, was given a wrong does of medication it might cost them their life. This indicates that there are always practical uses for mathematics in life as well as protective ones.

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Extended Essay- Math

    �{��gK -�-�i$�q�_ ßE�O�� �H��bX'������"�Q(r)5j"�1/2n�:���1O�~�i���(nl�5�"�K�$�%�>:w�l�s8\gx�e�L���@�.Ù·o��۷�=ztÛ¶m�d`@w�y' �(�(c)�0�@ *�*"$3/4<j8�?�G�u� Eb�C%[dj��YZ1/4x1��v����*3/4�-h�E���>b�c�Lmz5 "i'��^�q"�$9[�"�Q.<��ΥE)Re,R#Gx�é§N��;�1/4dh��� [qÇ7�x�yt��{��Ͳ1/2C0ТE����=�'�1/27n+�>}�=���L �$�c��...�< �A#��-��Í7�E��\"3/4��I[��0×���6����inB�� \<[�3���]�-%I��ٳ':EÒ (r)CL<�B_m�ZW5d�1/4o�E<-Y��Ν;�m�f���J���R��>�D�%�" �...�bz�\c�`2x/(c)��+����'��...0��-(1I�%�(Î9rΜ9]7n-J�5��% �0�d ���պR�<1 Q �w��]Tl��H�DM�8\j]ÖÚ� a]1��`'$(r)>��"),���-z�m�RX�"C��G�Y��1J�ҪU""�3/4"�(tm)Sxj�e8 ?���<��s�`r-�w�d]�HcRb\-� ����a�(r)8�c��|Q.�Zw�s��W\���gN1�o~�|A�$�7p��Æ9���� .��"if'���!2G:?��#D�4(^F'2���U�è¢ï¿½G�,��!�$���Up'o3/4�& 0-'���� 7v�)�\�vĪ�ִ�gkΜ n��S>��Ø���^��_��,á¡-Â��r^{�5*�P�0yJx��"-?�e...�ӹ(_WM �1�"̱�v*t�/��"(n�1/2{��~�m�"�;�-��%q6�'-9���%...t�B�X���"�M�R��"�Y�?�|3%jM'~��b �/�� t�A0M�#�H�p%�b51���bqqIrW�2M,X��o���� ��...i(tm)J(r)_��W%�Î$~S �)3/4`3/4)O ���Y�t�)���PY� ��&|�"p'�I=��<ꨣL�12�$ï¿½ï¿½Ï ï¿½1o���b"��_~?�;�~(n�y� ��~�9��*x9����?�$F�$B(r)�26l���O)b��((tm)�Y|Ô¨Q�G~�"����I�h1/4L�� 0��N#��q��t��1/4g�-� f�E� �&���kÒ¤I�>��a�V��-�a�֭�E BA?�'2�|�h���(tm)9�'�x���1/2�c�A �PiÖY�kb-LfΜi�B-1/4��s:1/2-T�S��g#�nM �1v ��O�]| �" |��g �xg� �O�'- ����7o-�,l�H3/4��"���g7'S[x���"��v$�`1/4+ :��"�6a^��3/4��FoN���Qd""3z� �q�֭c9&�"�Ce�m f �+�����l@_s����ܹs��{�C���"\q "obl�0a��� �m�С� r�s\�{�-M�1v�� ���O�|\����(r)K�U+�-��l 'S[�r �"�+xÜ|�l'�'�#������t��aO+_ �k�6�x�$'�(c)mzK����BÙ��ۿ5�dE��qÚr PNÈ£e��v\U�2�.

  2. Tide Modeling

    Once the comparison is made it is possible to see that the graph was moved 3 units to the right. Horizontal Translation c=3 Since the crest in the standard graph was at 0 and in the excel graph it is at 3 it is possible to assume it was moved 3 units to the right.

  1. IB Mathematics Portfolio - Modeling the amount of a drug in the bloodstream

    The only difference is the original data didn't have the line of best fit on it. It still contains the same data points and scale. The r2 or the coefficient of determination displays how closely my model fits the data.

  2. A logistic model

    The model considers an annual harvest of 1x104 fish. A population collapse occurs after year 25. From figure 7.2 one can observe the chronic depletion of the fish population as a consequence of excessive harvest. If one continues with the model into year 26 the population turns negative, and this means the death of the last fish in the lake.

  1. Math SL Fish Production IA

    a, b, c, d and e are: 3s.f Therefore when the values are substituted into the template equation, the equation can be rewritten as: Graph 7: This shows the function model for the 3rd section acquired from solving the matrix equation.

  2. Mathematics internal assessment type II- Fish production

    mathematical model found previously cannot be used for this set of data points. If this graph were to be plotted against the first graph, it would stay too close to the x-axis, thus not being able to utilize the model found earlier.

  1. Mathematic SL IA -Circles (scored 17 out of 20)

    It is not possible to draw three circles required. Through the 4 different types of test, we can notice that the general statement is valid on first three tests, but not for the last one. The only characteristic I found out was that the length of OP is shorter 2 units than the length of r.

  2. IB Math Methods SL: Internal Assessment on Gold Medal Heights

    Hence, the winning height in 2016 is rather unrealistic given the current situation. Let?s now expand our original data set from 1896 up until 2008. Shown below is the old data merged with the new information we have: Expanded Given Information Table Year 1 896 1 904 1 908 1

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work