MAT099 TASK 1: THE GROWTH OF BACTERIA TASK 2: TRAPEZOIDAL RULE LECTURER’S NAME: MISS NURASHIKIN BINTI ABDULLAH NO.
NAME
ID NUMBER
1.
ABU BAKAR BIN MUHAMMAD TAUFIK
2018295562
2.
AMMAR YUSOFF BIN FAISAL IBRAHIM
2018432744
3.
MOHAMMAD AMIRUL DANISH BIN FAIRUS
2018433052
4.
LUQMAN HAKEEM BIN SHAMSUL BAHARIN
2018633596
5.
SITI NURLIYANA BINTI HAMZANI
2018442524
6.
NUR MAISARAH BINTI NOR AZHARLUDIN
2018696484
MAT099
1.1 INTRODUCTION • For Task 1, the problem that we have to carry was to find the growth of bacteria. The data that we obtained is about the growth of mesophile bacteria as increase in time. These bacteria can easily divide in a lake or pond. The data is in second because the growth of this type of bacteria can increase in hot and moist surrounding very fast. • When calculate the theoretical value of the growth rate of bacteria using the formula that we had gained, there are differences when the theoretical data is obtained compared to the experimental data. The objective of this task is to compare the theoretical value and the experimental value of the growth of bacteria and to identified the factors that affects the growth of bacteria.
MAT099
DATA Time(sec)(t)
Number of Bacteria(B)
30
50 000
40
175 000
50
350 000
60
700 000
70
1 400 000
80
2 800 000
90
5 600 000
100
11 200 000
110
22 400 000
120
44 800 000
130
89 600 000
140
179 200 000
đ??ľ = đ??ľ đ?‘Ą denotes the number of bacteria at time t. At any time t, the rate of increase of the number of bacteria with respect to time is đ?‘‘đ??ľÎ¤đ?‘‘đ?‘Ą, so the assumption that rate of growth is proportional to the number of bacteria is described by differential equation. đ?‘‘đ??ľ âˆ?đ??ľ đ?‘‘đ?‘Ą đ?‘‘đ??ľ = đ?‘˜đ??ľ đ?‘‘đ?‘Ą đ?‘˜ = đ?‘?đ?‘œđ?‘›đ?‘ đ?‘Ąđ?‘Žđ?‘›đ?‘Ą đ?‘œđ?‘“ đ?‘?đ?‘&#x;đ?‘œđ?‘?đ?‘œđ?‘&#x;đ?‘Ąđ?‘–đ?‘œđ?‘›đ?‘Žđ?‘™
𝐿𝑒𝑡 𝐵 𝑡 : 𝑇ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑐𝑡𝑒𝑟𝑖𝑎 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡(𝑠𝑒𝑐𝑜𝑛𝑑) 175 000−50 000 • 𝑘1 = = 2.5 50 000
• 𝑘2 =
2 800 000−1 400 000 1 400 000
• 𝑘3 =
179 200 000−89 600 000 89 600 000
=1 =1
𝑘1 + 𝑘2 + 𝑘3 2.5 + 1 + 1 𝑘= = = 1.5 3 3 𝑑𝐵 ∴ = 1.5 ∙ 𝐵 𝑑𝑡 𝑠𝑒𝑝𝑎𝑟𝑎𝑡𝑒 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠, 1 ∙ 𝑑𝐵 = 1.5 ∙ 𝑑𝑡 𝐵
𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑡𝑒 𝑏𝑜𝑡ℎ 𝑠𝑖𝑑𝑒, 1 න 𝑑𝐵 = න 1.5 𝑑𝑡 𝐵 ln 𝐵 = 1.5𝑡 + 𝐶 𝐵 = 𝑒1.5𝑡+𝐶 𝐵 = 𝑒1.5𝑡 ∙ 𝑒 𝐶 𝐿𝑒𝑡 𝑒 𝐶 = 𝐴 ∴ 𝐵 = 𝐴𝑒1.5𝑡 => 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛
𝑊ℎ𝑒𝑛 𝑡 = 0(70), 𝐵 = 𝐵0 = 𝐴: 1 400 000 = 𝐴𝑒 0 𝐴 = 1 400 000 𝐵 = 1 400 000𝑒1.5𝑡 => 𝑝𝑎𝑟𝑡𝑖𝑐𝑢𝑙𝑎𝑟 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛
𝑷𝒓𝒆𝒅𝒊𝒄𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒕𝒘𝒐 𝒇𝒖𝒕𝒖𝒓𝒆
𝑷𝒓𝒆𝒅𝒊𝒄𝒕𝒊𝒐𝒏 𝒇𝒐𝒓 𝒕𝒘𝒐 𝒑𝒂𝒔𝒕
1. 𝑁𝑒𝑎𝑟 𝑓𝑢𝑡𝑢𝑟𝑒 𝑎𝑓𝑡𝑒𝑟 20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, 𝐵 = 1 400 000𝑒1.5(20) 𝐵 = 1.496 × 1019 𝑎𝑓𝑡𝑒𝑟 20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑐𝑡𝑒𝑟𝑖𝑎 𝑖𝑠 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 1.5 × 1019
1. 𝑁𝑒𝑎𝑟 𝑝𝑎𝑠𝑡 𝑏𝑒𝑓𝑜𝑟𝑒 20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝐵 = 1 400 000𝑒1.5(−20) 𝐵 = 1.310 × 10−7 𝑏𝑒𝑓𝑜𝑟𝑒 20 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑐𝑡𝑒𝑟𝑖𝑎 𝑖𝑠 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 1.3 × 10−7
2. 𝐷𝑖𝑠𝑡𝑎𝑛𝑡 𝑓𝑢𝑡𝑢𝑟𝑒 𝑎𝑓𝑡𝑒𝑟 40 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, 𝐵 = 1 400 000𝑒1.5(40) 𝐵 = 1.599 × 1032 𝑎𝑓𝑡𝑒𝑟 40 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑐𝑡𝑒𝑟𝑖𝑎 𝑖𝑠 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 1.6 × 1032
2. 𝐷𝑖𝑠𝑡𝑎𝑛𝑡 𝑝𝑎𝑠𝑡 𝑏𝑒𝑓𝑜𝑟𝑒 40 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 𝐵 = 1 400 000𝑒1.5(−40) 𝐵 = 1.226 × 10−20 𝑏𝑒𝑓𝑜𝑟𝑒 40 𝑠𝑒𝑐𝑜𝑛𝑑𝑠, 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑏𝑎𝑐𝑡𝑒𝑟𝑖𝑎 𝑖𝑠 𝑎𝑝𝑝𝑟𝑜𝑥𝑖𝑚𝑎𝑡𝑒𝑙𝑦 1.2 × 10−20
1.3 RESULT AND DISCUSSION What has affected the population growth in the past? Temperature Mesophiles are bacteria that grow best at moderate temperatures. Their optimum growth temperature is between 25C and 45C. Most bacteria are mesophilic and include common soil bacteria and bacteria that live in and on the body. pH The experiment that was conduct take place at pH level around 4 until 6. These are the optimum pH for increase the growth of mesophiles bacteria which is 4.6 until 6. As we know that mesophiles bacteria is acidophile which means it needs acidic surrounding for optimum growth.
Concentration of nutrient At the lake, there is a lot of nitrogen and phosphorus that produce by the plant at the lake. This is the main nutrient that can increase the growth of mesophiles bacteria. Other than that, the existence of oxygen that was produced by the green plant in the lake also contribute to the growth of bacteria Presence of water It is obvious that bacteria need water to survive. This is because it needs this kind of moisture surrounding to carry out the process of dividing. If its to hot and dry, the bacteria cannot undergo mitosis process and the bacteria would have died.
1.3 RESULT AND DISCUSSION What has affected the population growth in the future?
• Temperature increase • Presence of water higher than initial • Bacteria can easily increase in growth at warm and moist surrounding
MAT099
2.1 INTRODUCTION • In some equation, we cannot calculate using any basic integration that we had learned in the class. Other than that, we also cannot calculate some certain equation using by part, substitution and partial fraction so that is why we use trapezoidal rule. Trapezoidal rule is a technique for approximating the definite integral. The rule works by approximating the region under the graph of the function as a trapezoid and calculating its area.
TASK 2 QUESTION • Use the trapezium rule with (a) 5 ordinates (b) 9 4 ‍׏‏0 (3 + � � )dx,
ordinates to evaluate correct to four significant figures. Which one is more accurate, a) or b)? Why?
MAT099
5 ORDINATES
đ?‘Ľ
đ?‘“ đ?‘Ľ =3+đ?‘’
đ?‘Ľ
Coefficient
Coefficient∙ đ?‘“(đ?‘Ľ)
0
4.000
1
4.000
1
5.718
2
11.436
2
7.113
2
14.226
3
8.625
2
17.250
4
10.389
1
10.389
TOTAL
57.301
𝑓 𝑥 =3+𝑒
𝑥
CALCULATION 5 ORDINATES 𝑏
𝑨𝒓𝒆𝒂 𝒐𝒇 𝒓𝒆𝒈𝒊𝒐𝒏 = න 𝑓 𝑥 𝑑𝑥 ≈ 𝐴1 + 𝐴2 + 𝐴3 + 𝐴4 𝑎
ℎ ℎ ℎ 𝑦0 + 𝑦1 + 𝑦1 + 𝑦2 + 𝑦2 + 𝑦3 2 2 2 ℎ + 𝑦3 + 𝑦4 2 ℎ = 𝑦0 + 2𝑦1 + 2𝑦2 + 2𝑦3 + 𝑦4 2 =
𝑏
න 𝑓 𝑥 𝑑𝑥 ≈ 𝑎
ℎ ሾ𝑓 𝑎 + 2𝑓 𝑥1 + 2𝑓 𝑥2 + 2𝑓 𝑥3 2
9 ORDINATES đ?‘Ľ 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
đ?‘“ đ?‘Ľ =3+đ?‘’ 4.000 5.028 5.718 6.403 7.113 7.860 8.625 9.494 10.389 TOTAL
đ?‘Ľ
Coefficient 1 2 2 2 2 2 2 2 1
Coefficient∙ đ?‘“(đ?‘Ľ) 4.000 10.056 11.436 12.806 14.226 15.720 17.250 18.988 10.389 114.871
𝑓 𝑥 = 3+𝑒
CALCULATION 9 ORDINATES
𝑥
𝑨𝒓𝒆𝒂 𝒐𝒇 𝒓𝒆𝒈𝒊𝒐𝒏 𝑏
= න 𝑓 𝑥 𝑑𝑥 ≈ 𝐴1 + 𝐴2 + 𝐴3 + 𝐴4 + 𝐴5 + 𝐴6 + 𝐴7 + 𝐴8 𝑎
ℎ ℎ ℎ ℎ 𝑦0 + 𝑦0.5 + 𝑦0.5 + 𝑦1 + 𝑦1 + 𝑦1.5 + 𝑦1.5 + 𝑦2.0 2 2 2 2 ℎ ℎ ℎ ℎ + 𝑦2.0 + 𝑦2.5 + 𝑦2.5 + 𝑦3.0 + 𝑦3.0 + 𝑦3.5 + 𝑦3.5 + 𝑦4.0 2 2 2 2 =
𝑏
න 𝑓 𝑥 𝑑𝑥 ≈ 𝑎
ℎ ሾ𝑓 𝑎 + 2𝑓 𝑥0.5 + 2𝑓 𝑥1.0 + 2𝑓 𝑥1.5 + 2𝑓 𝑥2.0 2
2.3 RESULT AND DISCUSSION • The area of region between 0 to 4 with 9 ordinates is more precise compared to area of region between 0 to 4 with 5 ordinates. This is because the more subintervals we use, the less area that do not been cover by trapezium. • There are error in this trapezoidal rule. As the graph concave up, the error is negative while the graph concave down, the error is positive.
5 ORDINATES
9 ORDINATES
CONCLUSION • In conclusion, from this group project assignment the thing that we experience were we tend to work as a team and brainstorm on how to solve the problems that had been given to us together. Based on Task 1 that was given to us which is Growth, the growth of bacteria, the benefits that we gain was it increase our knowledge about growth problem that is related to mathematics and daily life. Next, based on Task 2, on certain equation we cannot calculate area under the graph using the method that we learn in our syllabus such as basic rule of integration, exponential function, integration by substitution, integration by parts and partial fraction method because of that from trapezoidal rule, we can calculate the area under the graph to get more precise value based on the more number of ordinates.