MATHEMATICAL the lower tips close to the nodes

    MATHEMATICAL INVESTIGATION STANDARD LEVEL TOPIC: Formula generation for thegrowth’s pattern of Basil (OcimumBasilicum) and Celery (Apium graveolens)plants’ cutting. By: NDZENGUE BABONG BETINA EYVRARDE MARIE CECILEFacilitator: Mister NGONGA DIVINE           TABLE OF CONTENTIntroduction………………………………………………………………………………..       3                                                                                                 Procedure…………………………………………………………………………………..      3 Finding the Basils’growth pattern’s equation…………………………………….

……….     4 Finding theCeleries’ growth pattern’s equation..

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      6 Verificationprocess………………………………………………………………………..   10 Conclusion and Evaluation…………………………………………………………………

    11 Referencing…………………………………………………………………………………

.   12            Introduction:Culinary herbs are indispensable ingredients in ourcountry’s food habits. In the contrary of common addible plants like beans orgroundnut, those herbs are not usually seeing in plantations or farms whilethey are highly consumed; that piqued my concerned to know how they grow. Whilesearching for my chemistry portfolio, I found a recent discovery on howculinary herbs can grow only with water as a stimulus.

In that discovery, youngshoots cuttings of the herbs are plunge into a container filled with water andleft in there for 1 to 6 weeks depending on the height you wish to attain. Thecomplexity of a scientific phenomenon not yet explained pushed me to look atthe mathematic behind it. More specifically the mathematical relationshipbetween the number of days in which the herbs are submerged and the averagegrowth rate of the herbs. The herbs I will use are: Basil (Ocimum Basilicum) andCelery (Apium graveolens) which are the ones readily available and commonlyused in my country. In order to find the equation for the pattern in the growthof those two plants, I will use scatter plot graphs obtained from the plottingof the value in tables containing information about the number of days takenand the change in length in each the basil and the celery.

Procedure:I used soft cuttings because they grow quickly inwater, so I won’t use any rooting or growing hormones. I grew herbs that I got directlyfrom the market. I washed them in plain water and cut off the lower portion. Afterthat, I removed the lower leaves from the cuttings and trim the lower tipsclose to the nodes from where the roots arise. Once they were inserted into thecontainers, I made sure there were any leaves touching the water: they could rooteasily and spoiled the water, as they usually do in flower vases. Because theroots generally like to grow away from light, I painted the bottom part of thecontainer in a dark color (dark brown) and I recovered the bottom still withbaking paper. To provide support the pant I put a plastic cooking paper on thetop of my containers. With a scaled beaker, I measure 55ml of water and I poured the same amount of it in four plasticcontainers (for the four different length of the plants).

I will measure thelength of the cuttings with a graduated ruler from the bottom tip to the apicalMeristem (the part where the bulb is found). The following tables show theinitial 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                                               Findingthe Basils’ growth pattern’s equationAfter the herbswere plunged in water for 1 week, I recorded the final length from the roots tothe tips. In the following table I will record the initial, final length andthe percentage in the Basils cuttings’ length; and the percentage change inlength will be calculated using the following formula:      Final length – InitiallengthTABLE 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 followingtable let x be the number of days used for each herbs and let y be the average changein 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 abovetable, the following graph can be plot, and from that graph I will derive the firstequation to determine the path of the growth of Basil cuttings. When I plot the graph, there is a positiveslope, meaning that an increase in the number of days leads to an increase inthe length of basils cuttings.

The shape of the graph is not really straightthus I have to find the best-fit line in order to generate the equation for thepattern of the growth. When I do so, I obtained the following graph.As indicated Ywith the spheres represent the initial graph and Y with no spheres representthe best-fit line. From the knowledge that a straight line is given by theequation y=mx+c, where m is the slope and c is the y-intercept, I am now going to find the different variablefor my equation take from. I will first start with the slope, knowing that to find it you can draw a right angle trianglewhere the intersection of the opposite side and the hypotenuse represent onepoint; and the intersection of the hypotenuse and the adjacent representanother 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 ywhen x=0. And according to table 1, when x=0, y=0.

Putting thevalues for our variables back in our formulas y=mx+c, I obtain y=0.95x+0but 0 being negligible the final equation is y=0.95x To verify myequation, I substitute any x value in the equation and look if the y value willbe approximately the same as the one on the graph as follows:X=8 will makey=0.95(8) = 7.6Looking closelyat the graph, we can see that the y value is 8 which is approximately the samething I calculated with my equation. There is a slight difference of 0.

4.Finding theCeleries’ growth pattern’s equationAfter the herbswere plunged in water for 1 week, I recorded the final length from the roots tothe tips. In the following table I will record the initial, final length andthe percentage in the Celeries cuttings’ length; and the percentage change inlength will be calculated using the following formula:      Final length – Initiallength 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 followingtable let x be the number of days used for each herbs and let y be the averagechange 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 abovetable, the following graph can be plot, and from that graph I will derive thefirst equation to determine the path of the growth of Celeries cuttings. In the abovegraph it can be depict that the slope is negative which means that as thenumber of days increased, the average length of the cuttings was decreasingresulting in the bending down of the graph. But it will be difficult to find anequation since the graph is undulating, thusly I will look for the beat-fitline as shown below.      The best-fitline here is represented by the line with no spheres on it and it can be seenthat the line is straight. As mentioned above, the equation will be on the formof y=mx+c.

I will use the alikeprocedure as for graph A. Slope (m): I will draw a right angle by linking the end of thebest-fit line with the end of the y-axis and the other end of the line with they-axis. Once the two lines intersect I will erase the left-overs.

One pointwill represent the intersection of the hypotenuse and the opposite side; andanother one will represent the intersection of hypotenuse and the adjacentside. This gave me the following.   As shown in thegraph, 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 previouslymentioned and it gives: .y-intercept: as done above, it is found by looking at the value ofy when x=0 and according to table B, when x=0, y=0.Having ourvariables, 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 theequation, I will use the alike verification for Basil cuttings’ equation. Itake x=6 and substitute it in the equation y=0.63(6) = -3.78. In the graph they value corresponding to x=6 is approximately 3.85 and there is only adifference of 0.

07. It can thusly be inferred that the equation is correct.           Verificationprocess: GRAPH A                                              GRAPH BWhen I plot my value from table 1 in my GraphingDisplay calculator (GDC) TI nspire-cx,I obtained approximately the same graph (graph A) that I previously got (graphB). Also looking at the equation the GDC calculated for the best-fit line y=0.

897427x+0.360964 is approximatelythe same thing with the one I calculated y=0.95x+0.The differences are only 0.06x and 0.36b.                     GRAPH C                                                             GRAPH DWhen I plot my value from table 4 in my GDC again, Iobtained approximately the same graph (graph C) that I previously got (graphD). Also looking at the equation the GDC calculated for the best-fit line y=-0.

637809x+0.316955 is approximatelythe same thing with the one I calculated y=-0.63x+0.The differences are only 0.008x and 0.32b.

  CONCLUSION ANDEVALUATIONThe complexity of a new study on culinary herbs pushedme to investigate on generating an equation that could determine the pattern ofthe 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 gathermy data for the initial length, the final length after each days, for 10consecutive 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 itand found my equation. I then used my GDC to make my verification the equationthe 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 manyreasons and uses. The main reason why I found the equations because it can beuseful to a more efficient and precise production of those commonly usedplants, and it will be easy to use that basic equations for those who can’thave huge plantations like housewives who wants to have an easy access to theseuseful herbs. The major use is that with that equation, cultivators of thoseplants will be able to estimate what quantities of hers they will be able toproduce 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 Iused by the desired quantities they will want to use and the number of herbsthey will want to grow; knowing that by growing those herbs they can be cut inmultiple parts and planted in the soil to reproduce many more.   Nonetheless, there were different possible errorswhich could have been the cause of the slight difference in the growing path’sequations during the verification process. The first one being when measuringthe increase in the lengths of the herbs, I could have read lengths slightlymore or less than what it really was. The second one being the rounding off ofthe calculation’s results that I did while making my analysis, it could havemake the final results not to be really accurate Lastly, the data recordingbecause to find the slope of the two equations I was approximating the readingof the coordinates for point A and point B. Those are the errors which couldhave made my different equations to be slightly inaccurate.

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