• MATHEMATICS AND THE WORLD AROUND US
REAL-LIFE, REAL-WORLD, CONTEX-BASED, APPLIED MATHEMATICS

### A play time - part I

The next tasks prepare you (or remind you) school tasks related to the functions and its graphs.

Do not worry about its difficulty!! If you successfully passed problems in the previous chapter ( it will be easy for you ;)

Medical students had a good time when they tryed a spirometry assessment at school:

Finn:

"I did not breath at the beginning of the assessment for 3 seconds!"

Matias:

"I inhaled deeply before we started so I had an opportunity to exhale deeply exactly at the beginning of the assessment!"

Sophi:

"I stopped breathing for a while after every inhalation!"

Hana:

"It is not possible to measure Vital Capacity based on my spirometry result!"

Find authors of the next spirometry curves. Correct answer is created by a pair: [ the first letter of the student's name + number of the spirometry curve].

Choose and remember the code (cipher) matched with your answer in a .

Answer F1, M2, S3, H4 F1, M4, S3, H2 F4, M3, S2, H1 F1, M2, S4, H3
code/cipher INEWTON GFBRIEMAN LEULER GWLEIBNIZ
###### task II: Truth - yes or no?

After the spirometry assessment our students (Finn, Matias, Sophi and Hana) got another task. They had to find volume of an air inhaled during their normal breathing.

After a while Sophi said: "It is very interesting! The result does not change if I move the graph of the spirometry curve across the paper. For example, like this!" (look on the right graph)

Is Sophi true?
What is the volume of an inhaled air in Sophi's example?

Answer Yes, Sophi is true. The volume of the normally inhaled air is approximately 600 ml No, Sophi is not true We do not know Yes, Sophi is true. The volume of the normally inhaled air is approximately 200 ml
code/cipher ... ... ... ...
###### task III: Frequency - the number of repetitions

The last task of our students was to find a breathing frequency (the number of repetitions of a normal inhalation and exhalation cycles in one minute). They worked with a spirometry curve drawn on the right picture.

Answer It is not possible to find the answer because we do not have the record of breading for the whole minute 12 20 40
code/cipher ... ... ... ...
###### task IV: A little bit of "pure" maths

The next task has two alternatives. Do you know a sinus function? If no, pick up the first alternative . If you already know it, the first alternative could not be interesting for you a from that reason we prepared the second alternative for you.

If you want, you can both variants! In this way you can check the accuracy of your your answer :).

###### alternative I

The right graph is the graph of a dependency between independent variable k and dependent variable m .

This dependency is in mathematics known like sinus . The graph is then graph of a sinus function .

Which of the next propositions are true for this graph?

A.

Dependent variable k only gets values of interval ⟨-1,1⟩.

B.

If independent variable m = π/2, the value of a dependent variable k is k = 1.

C.

If independent variable m ∈ ⟨π/2,3π/2⟩, its increasing values bring also increase of a k variable values.

D.

Values of a dependent variable k are repeating. The repetition distance is π (repetition frequency is π).

Answer A, B A, C B, C, D A, B, C, D
code/cipher ... ... ... ...
###### alternative II

The right graph is the graph of a dependency between independent variable k and dependent variable m .

Which of the next propositions are true for this graph?

A.

Dependent variable k only gets values of interval ⟨-1,1⟩.

B.

If independent variable m ∈ ⟨π,3π/2⟩, its increasing values bring also increase of a k variable values.

C.

If independent variable m = π/2, the value of a dependent variable k is k = 0.

D.

Values of a dependent variable k are repeating. The repetition distance is 3π/2 (repetition frequency is 3π/2).

Answer A, D B, C, D A, B, C, D C, D
code/cipher ... ... ... ...

PLEASE NOTE that the answers taken from the graph's data are only approximate! The difference between yours and our answers can be +-100ml.

Our patient's Forced Vital Capacity is approximately 2340ml.

If there is a big difference between yours and our answer, try to check the HELPER to this problem (bulb picture).

Yes, there is a difference.

Forced Vital Capacity of a healthy, 56 years old man who is 171cm tall should be 4.06l, but the patient's measured FVC is 2.34l.

Our patient's Forced Vital capacity is decreased.

There can be Restrictive lung disease expected but, to affirm this suspicion, it is necessary to know other parameters' values like Forced expiratory volume in 1 second (FEV1) or Peak expiratory flow (PEF).

(Missing parameters are discussed and can be discovered and understood in the next chapters.)

Our patient's Tidal Volume is approximately 0.5 liter.

Expiratory Reserve Volume is approximately 0.82 liter.

Inspiratory Reserve Volume is approximately 1.02 liter.

Note:There was a small change in an exhaled air at the end of the Spirometry - patient exhaled deeper. Mentioned change is caused by a forced exhalation in 9 - 11 second and we can neglect it in our counts.

Tidal volume of a healthy, 56 years old man, non-smoker who is 171 cm tall is lying in an interval <0.61 l ; 0.73 l> (15% - 18% of FVC of a healthy man, non-smoker, who has requested high and age).

Expiratory Reserve Volume of a healthy, 56 years old man, non-smoker who is 171 cm tall is 1.02 liter.

Inspiratory Reserve Volume of a healthy, 56 years old man, non-smoker who is 171 cm tall is 2.44 liter.

Our patient's measured values in comparison to the values of a healthy man, 56 years old, non-smoker, who is 171 cm tall.

56 YEARS OLD MAN, 171 cm FVC [l] TV [l] ERV [l] IRV [l]
MEASURED/PATIENT'S VALUES 2.34 0.5 0.82 1.02
IDEAL/TABLE VALUES 4.06 <0.6, 0.73> 1.02 2.44

something to think about: Is the y-location (location against the volume ax-axis) of a spirometry curve important? Is there any change on TV, IRV, ERV and FVC if we move it above/below?

#### Helpers

PLEASE NOTE that the answers taken from the graph's data are only approximate! The difference between yours and our answers can be +-100ml.

##### Helper 1.1

Firstly, it is necessary to understand the graph we are working with.

###### 1. Which Spirometry Curve parts are related to the patient's inhalations and which one to his exhalations?

To find an answer to this question, the imagine of a simple breathing can help. Try to close your eyes, inhale and exhale few times. Is the air volume in your lungs increasing or decreasing when you are inhaling air?

###### 2. When the patient's greatest exhalation started? (the exhalation during which patient exhaled the greatest volume of an air)

Note

If there are more than one of this type of exhalations, we are interested in the first one. The first one was executed on behalf of a doctor's appeal, after a forced inhalation.

###### 3. When the patient's greatest exhalation finished?

I am sure, you can solve the problem 1.1 easily now. What is the volume of an air exhaled during the greatest exhalation?

##### Helper 1.2

Try to look to the Forced Vital Capacity predicted values' table (a link is available after problem 1.2 assignment).

What Forced Vital Capacity should have a healthy man, who is 56 years old and 171cm high?

As a height unit, INCH is used in a table. Inches are usually used in an anglophone districts, most common in the USA and the United Kingdom (1 inch = 2.54 cm).

Man's Forced Vital Capacity Predicted Values
AGE (years)
HEIGHT (inches) 51 53 55 57 59 61 63 65
60 2.97 2.91 2.85 2.79 2.74 2.68 2.62 2.56
60.5 3.05 2.99 2.93 2.88 2.82 2.76 2.70 2.64
61 3.13 3.08 3.02 2.96 2.90 2.84 2.79 2.73
61.5 3.22 3.16 3.10 3.04 2.98 2.93 2.87 2.81
62 3.30 3.24 3.18 3.12 3.07 3.01 2.95 2.89
62.5 3.38 3.32 3.26 3.21 3.15 3.09 3.03 2.97
63 3.46 3.41 3.35 3.29 3.23 3.17 3.12 3.06
63.5 3.55 3.49 3.43 3.37 3.31 3.26 3.20 3.14
64 3.63 3.57 3.51 3.45 3.40 3.34 3.28 3.22
64.5 3.71 3.65 3.59 3.54 3.48 3.42 3.36 3.30
65 3.79 3.74 3.68 3.62 3.56 3.50 3.45 3.39
65.5 3.88 3.82 3.76 3.70 3.64 3.59 3.53 3.47
66 3.96 3.90 3.84 3.78 3.73 3.67 3.61 3.55
66.5 4.04 3.98 3.93 3.87 3.81 3.75 3.69 3.64
67 4.12 4.07 4.01 3.95 3.89 3.83 3.78 3.72
67.5 4.21 4.15 4.09 4.03 3.97 3.92 3.86 3.80
68 4.29 4.23 4.17 4.11 4.06 4.00 3.94 3.88
68.5 4.37 4.31 4.26 4.20 4.14 4.08 4.02 3.97
69 4.45 4.40 4.34 4.28 4.22 4.16 4.11 4.05
69.5 4.54 4.48 4.42 4.36 4.30 4.25 4.19 4.13
70 4.62 4.56 4.50 4.45 4.39 4.33 4.27 4.21
70.5 4.70 4.64 4.59 4.53 4.47 4.41 4.35 4.30
71 4.78 4.73 4.67 4.61 4.55 4.49 4.44 4.38
71.5 4.87 4.81 4.75 4.69 4.63 4.58 4.52 4.46
72 4.95 4.89 4.83 4.78 4.72 4.66 4.60 4.54
72.5 5.03 4.97 4.92 4.86 4.80 4.74 4.68 4.63
73 5.11 5.06 5.00 4.94 4.88 4.82 4.77 4.71
73.5 5.20 5.14 5.08 5.02 4.96 4.91 4.85 4.79
74 5.28 5.22 5.16 5.11 5.05 4.99 4.93 4.87
74.5 5.36 5.30 5.25 5.19 5.13 5.07 5.01 4.96
75 5.44 5.39 5.33 5.27 5.21 5.15 5.10 5.04
75.5 5.53 5.47 5.41 5.35 5.30 5.24 5.18 5.12
76 5.61 5.55 5.49 5.44 5.38 5.32 5.26 5.20
76.5 5.69 5.63 5.58 5.52 5.46 5.40 5.34 5.29
77 5.77 5.72 5.66 5.60 5.54 5.48 5.43 5.37
77.5 5.86 5.80 5.74 5.68 5.63 5.57 5.51 5.45
78 5.94 5.88 5.82 5.77 5.71 5.65 5.59 5.53
78.5 6.02 5.96 5.91 5.85 5.79 5.73 5.67 5.62
79 6.10 6.05 5.99 5.93 5.87 5.81 5.76 5.70
79.5 6.19 6.13 6.07 6.01 5.96 5.90 5.84 5.78
80 6.27 6.21 6.15 6.10 6.04 5.98 5.92 5.86
80.5 6.35 6.29 6.24 6.18 6.12 6.06 6.00 5.95
81 6.44 6.38 6.32 6.26 6.20 6.15 6.09 6.03
81.5 6.52 6.46 6.40 6.34 6.29 6.23 6.17 6.11
82 6.60 6.54 6.48 6.43 6.37 6.31 6.25 6.19
82.5 6.68 6.62 6.57 6.51 6.45 6.39 6.33 6.28
83 6.77 6.71 6.65 6.59 6.53 6.48 6.42 6.36
83.5 6.85 6.79 6.73 6.67 6.62 6.56 6.50 6.44
84 6.93 6.87 6.81 6.76 6.70 6.64 6.58 6.52
84.5 7.01 6.95 6.90 6.84 6.78 6.72 6.66 6.61
85 7.10 7.04 6.98 6.92 6.86 6.81 6.75 6.69
##### Helper 1.3

Try to look to the Pulmonary Diseases' table (a link is available after problem 1.3 assignment).

What kind of a disease is signalized by a reduced Forced Vital Capacity?

##### Helper 1.4

Try to go back to the text before the problem 1.4, read it carefully again and pay attention to the graph.

What exactly are describing concepts Tidal Volume, Expiratory Reserve Volume, Inspiratory Reserve Volume?

Which parts of our patient's Spirometry curve represent these volumes?

##### Helper 1.5

Try to go back to the text before the problem 1.4 and read it carefully once again.

What Vital capacity should have a healthy man who is 56 years old and 171 centimeters tall?

What percentage of a Vital Capacity should be reached in Tidal Volume for a healthy human?

What percentage of a Vital Capacity should be reached in Expiratory and Inspiratory Reserve Volume?

##### Helper 1.6

What Vital capacity should have a healthy man who is 56 years old and 171 centimeters tall?

What Tidal Volume, Expiratory Reserve Volume and Inspiratory Reserve Volume should he has?

Try to put these volumes to our patient's spirometry curve?

#### Forum

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