MATHEMATICAL EYVRARDE MARIE CECILE Facilitator: Mister NGONGA

MATHEMATICAL INVESTIGATION STANDARD LEVEL

 

TOPIC: Formula generation for the
growth’s pattern of Basil (Ocimum
Basilicum) and Celery (Apium graveolens)
plants’ cutting.

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By: NDZENGUE BABONG BETINA EYVRARDE MARIE CECILE

Facilitator: Mister NGONGA DIVINE

 

 

 

 

 

 

 

 

 

 

 

TABLE OF CONTENT

Introduction………………………………………………………………………………..       3

                                                                                                

Procedure…………………………………………………………………………………..      3

 

Finding the Basils’
growth pattern’s equation…………………………………….………. 
    4

 

Finding the
Celeries’ growth pattern’s equation…………………………………………….      6

 

Verification
process………………………………………………………………………..   
10

 

Conclusion and Evaluation…………………………………………………………………    11

 

Referencing………………………………………………………………………………….   12

 

 

 

 

 

 

 

 

 

 

 

 

Introduction:

Culinary herbs are indispensable ingredients in our
country’s food habits. In the contrary of common addible plants like beans or
groundnut, those herbs are not usually seeing in plantations or farms while
they are highly consumed; that piqued my concerned to know how they grow. While
searching for my chemistry portfolio, I found a recent discovery on how
culinary herbs can grow only with water as a stimulus. In that discovery, young
shoots cuttings of the herbs are plunge into a container filled with water and
left in there for 1 to 6 weeks depending on the height you wish to attain. The
complexity of a scientific phenomenon not yet explained pushed me to look at
the mathematic behind it. More specifically the mathematical relationship
between the number of days in which the herbs are submerged and the average
growth rate of the herbs. The herbs I will use are: Basil (Ocimum Basilicum) and
Celery (Apium graveolens) which are the ones readily available and commonly
used in my country. In order to find the equation for the pattern in the growth
of those two plants, I will use scatter plot graphs obtained from the plotting
of the value in tables containing information about the number of days taken
and the change in length in each the basil and the celery.

Procedure:

I used soft cuttings because they grow quickly in
water, so I won’t use any rooting or growing hormones. I grew herbs that I got directly
from the market. I washed them in plain water and cut off the lower portion. After
that, I removed the lower leaves from the cuttings and trim the lower tips
close to the nodes from where the roots arise. Once they were inserted into the
containers, I made sure there were any leaves touching the water: they could root
easily and spoiled the water, as they usually do in flower vases. Because the
roots generally like to grow away from light, I painted the bottom part of the
container in a dark color (dark brown) and I recovered the bottom still with
baking paper. To provide support the pant I put a plastic cooking paper on the
top of my containers. With a scaled beaker, I measure 55ml of water and I poured the same amount of it in four plastic
containers (for the four different length of the plants). I will measure the
length of the cuttings with a graduated ruler from the bottom tip to the apical
Meristem (the part where the bulb is found). The following tables show the
initial length of Basil and Celery plants cuttings.

TABLE A: Basil cuttings’ information

Plant reference

Initial Length (cm)

B1

18.5

B2

19.2

B3

19.7

 TABLE B: Celery cuttings’ information

Plant reference

Initial Length (cm)

C1

16.5

C2

22.3

C3

13.5

 

                                             Finding
the Basils’ growth pattern’s equation

After the herbs
were plunged in water for 1 week, I recorded the final length from the roots to
the tips. In the following table I will record the initial, final length and
the percentage in the Basils cuttings’ length; and the percentage change in
length will be calculated using the following formula:      Final length – Initial
length

TABLE 1:

B1

B2

B3

Days

Initial
length
 

Final
length
 

Change
in
length
 

Days

Initial
length
 

Final
length
 

Change          in
length
 

Days

Initial
length
 

Final
length
 

Change
in
length

0

18.5

18.5

0

0

19.2

19.2

0

0

19.7

19.7

0

1

18.5

19.5

1

1

19.2

20.3

1.1

1

19.7

21.1

1.4

2

18.5

20.9

2.4

2

19.2

21.7

2.5

2

19.7

22.5

2.8

3

18.5

21.8

3.3

3

19.2

22.5

3.3

3

19.7

23.4

3.7

4

18.5

22.6

4.1

4

19.2

23.6

4.4

4

19.7

24.2

4.5

5

18.5

23.7

5.2

5

19.2

24.5

5.3

5

19.7

25.6

5.9

6

18.5

24.5

6

6

19.2

25.1

5.9

6

19.7

26.2

6.5

7

18.5

25.6

7.1

7

19.2

26.2

7

7

19.7

27.1

7.4

8

18.5

26.8

8.3

8

19.2

26.4

7.2

8

19.7

27.5

7.8

9

18.5

27

8.5

9

19.2

26.9

7.7

9

19.7

27.9

8.2

10

18.5

27.4

8.9

10

19.2

30.1

10.9

10

19.7

30.2

10.5

 

In the following
table let x be the number of days used for each herbs and let y be the average change
in length of the herbs which is finds using the following formula :.  

TABLE 2:

X

y

0

0

1

1.16

2

2.566667

3

3.433333

4

4.333333

5

5.466667

6

6.133333

7

7.166667

8

7.766667

9

8.133333

10

10.1

 

From the above
table, the following graph can be plot, and from that graph I will derive the first
equation to determine the path of the growth of Basil cuttings.

 When I plot the graph, there is a positive
slope, meaning that an increase in the number of days leads to an increase in
the length of basils cuttings. The shape of the graph is not really straight
thus I have to find the best-fit line in order to generate the equation for the
pattern of the growth. When I do so, I obtained the following graph.

As indicated Y
with the spheres represent the initial graph and Y with no spheres represent
the best-fit line. From the knowledge that a straight line is given by the
equation y=mx+c, where m is the slope and c is the y-intercept, I am now going to find the different variable
for my equation take from. I will first start with the slope, knowing that to find it you can draw a right angle triangle
where the intersection of the opposite side and the hypotenuse represent one
point; and the intersection of the hypotenuse and the adjacent represent
another one. This is what I did as follows

 

As shown above,
there are point A and B with approximate coordinates A (10; 9.5) and B (0; 0.5)
and I just insert the values in the formula  . It gives .

For the y-intercept, known as the value of y
when x=0. And according to table 1, when x=0, y=0.

Putting the
values for our variables back in our formulas y=mx+c, I obtain y=0.95x+0
but 0 being negligible the final equation is y=0.95x

To verify my
equation, I substitute any x value in the equation and look if the y value will
be approximately the same as the one on the graph as follows:

X=8 will make
y=0.95(8) = 7.6

Looking closely
at the graph, we can see that the y value is 8 which is approximately the same
thing I calculated with my equation. There is a slight difference of 0.4.

Finding the
Celeries’ growth pattern’s equation

After the herbs
were plunged in water for 1 week, I recorded the final length from the roots to
the tips. In the following table I will record the initial, final length and
the percentage in the Celeries cuttings’ length; and the percentage change in
length will be calculated using the following formula:      Final length – Initial
length

TABLE 3:

C1

C2

C3

DAYS

Initial
length
 

Final
length
 

Change
    In length
 

Days

Initial
length
 

Final
length
 

Change          in
length
 

Days

Initial
length
 

Final
length
 

Change
   In
length

0

16.5

16.5

0

0

22.3

22.3

0

0

13.5

13.5

0

1

16.5

16

-0.5

1

22.3

22.0

-0.3

1

13.5

13.2

-0.3

2

16.5

15.8

-0.7

2

22.3

21.8

-0.5

2

13.5

13

-0.5

3

16.5

14.6

-1.9

3

22.3

20.9

-1.4

3

13.5

12.8

-0.7

4

16.5

13.9

-2.6

4

22.3

19.8

-2.5

4

13.5

12.1

-1.4

5

16.5

13.1

-3.4

5

22.3

19.1

-3.2

5

13.5

10.7

-2.8

6

16.5

12.7

-3.8

6

22.3

18.5

-3.8

6

13.5

10.2

-3.3

7

16.5

12.1

-4.4

7

22.3

18

-4.3

7

13.5

9.6

-3.9

8

16.5

11.6

-4.9

8

22.3

17.7

-4.6

8

13.5

8.9

-4.6

9

16.5

10.5

-6

9

22.3

17.3

-5

9

13.5

8

-5.5

10

16.5

9.9

-6.6

10

22.3

16.9

-5.4

10

13.5

7.5

-6

 

In the following
table let x be the number of days used for each herbs and let y be the average
change in length of the herbs, which is finds using the following formula:.    

TABLE 4:

X

y

0

0

1

-0.36

2

-0.56667

3

-1.33333

4

-2.16667

5

-3.13333

6

-3.63333

7

-4.2

8

-4.7

9

-5.5

10

-6

 

From the above
table, the following graph can be plot, and from that graph I will derive the
first equation to determine the path of the growth of Celeries cuttings.

 

In the above
graph it can be depict that the slope is negative which means that as the
number of days increased, the average length of the cuttings was decreasing
resulting in the bending down of the graph. But it will be difficult to find an
equation since the graph is undulating, thusly I will look for the beat-fit
line as shown below.

    

The best-fit
line here is represented by the line with no spheres on it and it can be seen
that the line is straight. As mentioned above, the equation will be on the form
of y=mx+c. I will use the alike
procedure as for graph A.

Slope (m): I will draw a right angle by linking the end of the
best-fit line with the end of the y-axis and the other end of the line with the
y-axis. Once the two lines intersect I will erase the left-overs. One point
will represent the intersection of the hypotenuse and the opposite side; and
another one will represent the intersection of hypotenuse and the adjacent
side. This gave me the following. 

 

As shown in the
graph, there are point A and B with approximate coordinates A (0; 0.3) and B
(10; -6). I insert those values in the formula of the slope previously
mentioned and it gives: .

y-intercept: as done above, it is found by looking at the value of
y when x=0 and according to table B, when x=0, y=0.

Having our
variables, I replace them in the equation mentioned and that gives: y=-0.63x+0. But 0 being insignificant,
the final equation is y=-0.63x.

To verify the
equation, I will use the alike verification for Basil cuttings’ equation. I
take x=6 and substitute it in the equation y=0.63(6) = -3.78. In the graph the
y value corresponding to x=6 is approximately 3.85 and there is only a
difference of 0.07. It can thusly be inferred that the equation is correct.

 

 

 

 

 

 

 

 

 

 

Verification
process:

 

GRAPH A                                              GRAPH B

When I plot my value from table 1 in my Graphing
Display calculator (GDC) TI nspire-cx,
I obtained approximately the same graph (graph A) that I previously got (graph
B). Also looking at the equation the GDC calculated for the best-fit line y=0.897427x+0.360964 is approximately
the same thing with the one I calculated y=0.95x+0.
The differences are only 0.06x and 0.36b.

 

                   GRAPH C                                                             
GRAPH D

When I plot my value from table 4 in my GDC again, I
obtained approximately the same graph (graph C) that I previously got (graph
D). Also looking at the equation the GDC calculated for the best-fit line y=-0.637809x+0.316955 is approximately
the same thing with the one I calculated y=-0.63x+0.
The differences are only 0.008x and 0.32b.

 

 

CONCLUSION AND
EVALUATION

The complexity of a new study on culinary herbs pushed
me to investigate on generating an equation that could determine the pattern of
the growth of the most consumed herbs in my country which are Basil and Celery.
Following the similar procedure as the study I red online, I succeed to gather
my data for the initial length, the final length after each days, for 10
consecutive days. I did it with the two plants separately. With the data I got,
I plotted the graph in Microsoft office EXCEL 2016 and I started my analysis.
Once I found the way through which I could find my equation I went through it
and found my equation. I then used my GDC to make my verification the equation
the GDC calculated was approximately the same thing as the one I calculated.
The finding of the equations of the growth’s pattern of those plants have many
reasons and uses. The main reason why I found the equations because it can be
useful to a more efficient and precise production of those commonly used
plants, and it will be easy to use that basic equations for those who can’t
have huge plantations like housewives who wants to have an easy access to these
useful herbs. The major use is that with that equation, cultivators of those
plants will be able to estimate what quantities of hers they will be able to
produce according to a particular amount of water for a precise number of days.
They will be able to do so by simply multiplying the different quantities I
used by the desired quantities they will want to use and the number of herbs
they will want to grow; knowing that by growing those herbs they can be cut in
multiple parts and planted in the soil to reproduce many more.   

Nonetheless, there were different possible errors
which could have been the cause of the slight difference in the growing path’s
equations during the verification process. The first one being when measuring
the increase in the lengths of the herbs, I could have read lengths slightly
more or less than what it really was. The second one being the rounding off of
the calculation’s results that I did while making my analysis, it could have
make the final results not to be really accurate Lastly, the data recording
because to find the slope of the two equations I was approximating the reading
of the coordinates for point A and point B. Those are the errors which could
have made my different equations to be slightly inaccurate.

 

 

 

 

REFERENCING:

 

·        
Natural
living ideas.(2017). Natural living
ideas’ website. Retrieved 22 October, 2017, from http://www.naturallivingideas.com/herbs-vergetables-plants-yo-grow-in-water/