add-maths project 2015

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ADDITIONAL MATHEMATICS PROJECT WORK 2015

TITLE: COFFEENAME: SCHOOL NAME: IC NUMBER: ADVISOR:

Content Page

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1Title3

2Introduction4

3Task 11(a) Draw the graph of y = x2-2 for 0 x 2 (b) Find the expression of V in terms of y using Integration (c) Find the time taken to fill up the cup with coffee

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4Task 22(a) Find the change of coffee in the cup, V with respect to the height, y of the coffee level in the cup (b) (i) Determine and tabulate the rate of increase of the height of the coffee in the cup (ii) Interpretation of data and discussion

6,7

5Task 3 The Investigation of the efficiency of the new mechanism 3(a) Determine and tabulate the rate of increase of the height of the coffee in the cup (b) (i) Draw a Scatter Plot using Microsoft Excel to compare the results with those in 2(b) (ii) Give comments (c) (i) Calculate the time taken to fill up the coffee cup with the new mechanism (ii) Interpretation of data and discussion 8,9

6Task 4 The investigation of the rate of flow of coffee in the cup when k varies 4(a) Determine and tabulate the rate of flow of coffee into the cup when k varies using Microsoft Excel (b) Determine the time taken to fill up the cup using Microsoft Excel 9,10

7ConclusionDetermine the best rate of flow of coffee into the cup.Discuss whether the new mechanism is effective or not

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8Further Exploration : Coffee It is good or bad for us?(a) List out 5 reasons for the positive effects and negative effects of drinking coffee(b) Use i-Think map to represent answer in (a)11,12,13

9Reflection(a) What have I learned while doing the Add. Maths Project Work?(b) Moral values that I practice when doing the project14

COFFEE

INTRODUCTION

coffee,a tree, its seeds, and the beverage made from them. The coffee tree, a small evergreen of the genusCoffea,has smooth, ovate leaves and clusters of fragrant white flowers that mature into deep red fruits about 1/2 in. (1.27 cm) long. The fruit usually contains two seeds, the coffee beans.C. arabicayields the highest-quality beans and provides the bulk of the world's coffee, including c.80% of the coffee imported into the United States. The species is thought to be native to Ethiopia, where it was known before A.D. 1000.Coffee's earliest human use may have been as a food; a ball of the crushed fruit molded with fat was a day's ration for certain African nomads. Later, wine was made from the fermented husks and pulps. Coffee was known in 15th-century Arabia; from there it spread to Egypt and Turkey, overcoming religious and political opposition to become popular among Arabs. At first proscribed by Italian churchmen as a heathen's drink, it was approved by Pope Clement VIII, and by the mid-17th cent. coffee had reached most of Europe. Introduced in North America c.1668, coffee became a favorite American beverage after theBoston Tea Partymade tea unfashionable.Coffee owes its popularity in part to the stimulative effect of itscaffeineconstituent. Caffeine, a bitter alkaloid, can also contribute to irritability, depression, diarrhea, insomnia, and other disorders. Decaffeinated coffees, developed in the early 1900s, account for c.18% of the U.S. market. For those without the time or the inclination to brew their own, there are instant or soluble coffees, introduced in 1867, which account for c.17% of U.S. coffee sales.

Task 11(a) Draw the graph of y = x2-2 for 0 x 2Method: 1. Calculate the value of y which 0 x 2.2. Draw a table to represent the data.3. Using the table data, draw a graph using a graph paper.4. Let the vertical line as y-axis and the horizontal be the x-axis.5. Plot the graph using the table.6. Draw the line of the graph.

x012

y-2-12

Table 1.1A parabola curve-shaped graph is formed.

(b) Find the expression of V in terms of y using Integration.

Method.1. Use the formula y =x2-2 to find the expression of the volume of a cup of coffee by using in integration method.

Working:v=y x dy=y y+2 dy=[(y/2)+2y]y=[(y/2)+2y]-[(0/2)+2(0)]=[(y/2)+2y]

Therefore, the expression of V in terms of y using the integration method is V=[(y/2)+2y].

(c) Find the time taken to fill up the cup with coffee

Method:1. By using the expression found in 1(b) , V=[(y/2)+2y],calculate the volume of the cup.2. Substitute y = 7 to get the value of volume of cup.3. Find the rate of volume of the cup by using differentiation method.\4. Let the differentiated expression equals to 5.5. Calculate the time taken to fill up the coffee.

Working:v=x dy= y+2 dy=(y/2)+2y ]=[(7/2)+2(7)]-[(0/2)+2(0)]=38.5unit

V=38.5(dv/dt)=5(38.5/t)=5t= (38.5/5)t=24.19 seconds Therefore, the time taken for the coffee to fill up the cup of coffee is 24.19 seconds.Task 22(a) Find the change of coffee in the cup, V with respect to the height, y of the coffee level in the cupMethod:1. By using differentiation method, the change of coffee in the cup is found.

Working:(dV/dh)=(2(y/2) =(y+2)

Therefore, the change of coffee in the cup, V with respect to the height, y of the coffee level in the cup is dV/ dh==(y+2)

(b) (i) Determine and tabulate the rate of increase of the height of the coffee in the cup

Method:1. By using differentiation method, the rate of increase of the height of the coffee in the cup ,dh/dt is found.2. A table is drawn to tabulate the data.

Working:(dy/dt)=(dy/dv)*(dy/dt)(dv/dt)=5cms-1

ydy/dt

00.795774715

10.530516477

20.397887358

30.318309886

40.265258238

50.227364204

60.198943679

70.176838826

Table 2.1

Therefore, the data is filled with the value of the change of coffee in the cup, V with respect to the height, y of the coffee level in the cup and the rate of increase of the height of the coffee in the cup.

(ii) Interpretation of data and discussion

Method :1. A graph is drawn to interpret the data. 2. Let the vertical axis be the dy/dt and the horizontal axis be the y. 3. Plot the graph using the table above. 4. A line is drawn to connect the points.5. Analyze the graph.

From the graph, I know that the highest rate of the graph is when the y equals to 0 and the dy/dt equals to 0.8cm3s-1. The graph also shows the decreasing of the rate of increase of the height of the coffee in the cup. The lowest rate of graph is 0.19cm3s-1. The rate of increase of the height of the coffee is considered as low. When the height ,y increases, the rate of increases in the height of coffee decreases.This is because when the coffee fill up in the cup from the bottom to the top,the rate should reduce to prevents the coffee spill out of the cup.

Task 3 The Investigation of the efficiency of the new mechanism

3(a) Determine and tabulate the rate of increase of the height of the coffee in the cup

Method:1. By using differentiation method, a new expression of the rate of increase of the height of the coffee of a cup is found.

Working (dy/dt)=(dy/dv)*(dv/dt)=[1/(2+y)]*(10/y+1)=10/(y+1)(y+2)

Therefore the equation is found as dy/dt =10/(y+1)(y+2).

ydy/dt

01.591549431

10.530516477

20.265258238

30.159154943

40.106103295

50.075788068

60.056841051

70.044209706

Table 3.1 (b) (i) Draw a Scatter Plot using Microsoft Excel to compare the results with those in 2(b) Method :1. By using the Microsoft Excel, a graph is drawn.2. Let the vertical axis be the rate of increase of the height of the cup of coffee and the horizontal axis be the y.3. Plot the graph using the table of table above.4. Draw a line to connect all the points5. Label the x-axis and the y-axis and the title.

Working

(ii) Give comments

From the graph above, it shows the new dy/dt is higher than the old dy/dt at first. But as the value of y=1, the graph of the old dy/dt meets the new dy/dt. The new dy/dt is started to lower than the old dy/dt. It shows that the new machine of dy/dt is not successful built. The rate of increase of the value of the height of the cup of new machine is still below of the old machine. Improvement still need to be done so that the rate can be higher.

(c) (i) Calculate the time taken to fill up the coffee cup with the new mechanism

Method:1. By using the b) T= (dt/dy)*dy, the time taken is been calculated.

WorkingT= (dt/dy)*dy= (y+1)(y+2)/10 dy= (y+3y+2)/10 dy=(1/10) [(y/3)+(3y/2)+2y]=(1/10) [(7/3)+(3(7)/2)+2(7)] =63.41 seconds

Therefore, the time taken for the coffee to fill up the cup is 63.41 seconds.

(ii) Interpretation of data and discussionThe time taken for the coffee to fill up the cup of a new mechanism is longer than the old mechanism. Therefore, the new built of new mechanism is still need to be improve.

Task 4 The investigation of the rate of flow of coffee in the cup when k varies 4(a) Determine and tabulate the rate of flow of coffee into the cup when k varies using Microsoft Excel

Method :1. By using integration method, the expression of new mechanism of the rate of increase of the level of the cup is calculated.

Working:T =dt/dy dy= _0^7((y+2)(y+1))/k dy=/k _0^7(y+2)(y+1) dy=/k _0^7y^2+3y+2 dy=/k[y^3/3+(3y^2)/2+2y]70=/k(201.83)

Therefore, the expression is found as T= /k(201.83) (b) Determine the time taken to fill up the cup using Microsoft Excel

From the graph above, I found that the new mechanism is successfully built. The new mechanism has a higher rate of increase of the level of the coffee cup. Therefore, the new coffee dispensing machine is useable and more convenientConclusionDetermine the best rate of flow of coffee into the cup. Discuss whether the new mechanism is effective or not.The best rate of flow of coffee into the cup is the third coffee dispensing machine which has the highest rate of 63.25cm3s-1. Therefore, the coffee cup can be filled up is faster in one minute. The third mechanism is effective.

Further Exploration : Coffee It is good or bad for us? (a) List out 5 reasons for the positive effects and negative effects of drinking coffee (b) Use i-Think map to represent answer in (a)

Further Exploration(a) Positive effects of drinking coffee: Coffee can lower risk of some cancer. Some substances in coffee have anti-inflammatory and antioxidant effects, which benefit in chemoprevention activity besides influence cancer-associated genes.Drinking coffee can help to fight depression. Coffee contains large amounts of caffeine which is a well-known brain stimulant.Caffeine and its major metabolites may lower risk of depression among coffee drinkers. Liver can be protected by drinking coffee. Coffee consumption reduces DNA damage in blood cells and prevents ex vivo-induced DNA damage in healthy volunteers. Substances in coffee help to burn fat. Caffeine in coffee help mobilize fats from the fat tissues and increase metabolism. Chlorogenic acid help slow absorption of carbohydrates. Coffee will increase energy levels. Caffeine in coffee quickly stimulates your central nervous system, reduces your physical fatigue, and restores alertness when you feel drowsy.

Negative effects of drinking coffee:Drinking coffee can cause indigestion. Coffee increases stomach acid production, which could lead to indigestion as a normal side effect. Coffee would lead to headaches. When someone regularly drinks coffee, the body adjusts in essence fighting this effect. When coffee isn't consumed the result is that blood vessels dilate too much, which causes a headache. Blood pressure will increase by drinking coffee. Coffee quickly increased the stiffness of the large arteries in people. Coffee drinking would also lead to insomnia. Coffee cannot replace sleep, it can temporarily make us feel more alert by blocking sleep-inducing chemicals in the brain and increasing adrenaline production. Coffee can causes one to increase anxiety. Coffee can lead to a feeling of being jittery and an increased heart rate.(b)i-THINK map

Reflection(a) What have I learned while doing the Add. Maths Project Work?(b) Moral values that I practice when doing the projectREFLECTIONI learnt to calculate the values using integration and differentiation. Besides, I also learnt to draw graph using computer. I now know the facts of coffee while doing the Additional Mathematics Project Work. From this project, I have learnt to calculate the rate of dispensing of one coffee dispensing machine.

Coffee is good for us if we drink it properly. We should not drink it when we are in hungry condition. Although coffee can reduce our rate of having type II diabetes but it increases our rate of having heart disease and stroke problems. Therefore, we should not overdrink coffee as caffeine is not good for our health in long term.

There are several moral values that I have practised when doing this project. I have learnt to be determined and never give up when failing getting the results and difficulties. I also learn to be independence when I doing this project. I have tried to do this project without other people help. I also be thankful when my teacher and my parents to encourage me when I almost feel want to give up but I still continue to do this project.