Math Portfolio
Math Portfolio
HL- Type 1
INVESTIGATINGRATIOS OF AREAS AND VOLUMES
The purpose of this portfolio is to investigate the ratios of areas and volumes when a function y= xn is graphed between two arbitrary parameters x=a and x=b such that a‹b.
Task 1
The general formula to find area A is [pic]
The general formula to find area B is [pic]
Therefore, the ratio of Area A to Area B is-
= [pic] ÷ [pic]
= [pic] × [pic]
= n : 1
n:1 is the general conjecture formed.
The given function is in the form of y=xn. The function is y=x2. As mentioned above the parameters are between x=a and x=b. Here a=0 and b=1.
The graph of the function-
[pic]
y=xn where n=2.
The Area of the shaded region or area B was found to be 0.3333333333 units2.
Area A = 1- Area B = 1- 0.33333333 = 0.6666666666 units2
Mathematically the area can also be found by the following-
Area A can be found by the formula- [pic]
Area B can be found by the formula- [pic]
Total Area = 1 unit2
Area B = [pic]
= [pic]
Area A = [pic]
= [pic]
Ratio of area A : area B
[pic] : [pic]
= 2 : 1(satisfies the conjecture n:1)
Now I will investigate the ratios of area A and B with other natural numbers.
This means that the function will stay y=xn but here only n will change. The parameters will stay the same a=0 and b=1.
The ratios when n =3
The graph of y=x3
[pic]
Y=xn where n=3.
The Area of the shaded region or area B was found to be 0.25 units2.
Area A = 1- Area B = 1- 0.25 = 0.75 units2
Mathematically the area can also be found by the following-
Total area = 1unit2
Area B = [pic]
= [pic]
Area A = [pic]
= [pic]
Ratio of area A : area B
[pic] : [pic]
= 3 : 1(satisfies the conjecture n:1)
Now I will investigate the ratio of the areas for n being a rational number.
The ratio when n= [pic]
Y = x0.5
[pic]
HL- Type 1
INVESTIGATINGRATIOS OF AREAS AND VOLUMES
The purpose of this portfolio is to investigate the ratios of areas and volumes when a function y= xn is graphed between two arbitrary parameters x=a and x=b such that a‹b.
Task 1
The general formula to find area A is [pic]
The general formula to find area B is [pic]
Therefore, the ratio of Area A to Area B is-
= [pic] ÷ [pic]
= [pic] × [pic]
= n : 1
n:1 is the general conjecture formed.
The given function is in the form of y=xn. The function is y=x2. As mentioned above the parameters are between x=a and x=b. Here a=0 and b=1.
The graph of the function-
[pic]
y=xn where n=2.
The Area of the shaded region or area B was found to be 0.3333333333 units2.
Area A = 1- Area B = 1- 0.33333333 = 0.6666666666 units2
Mathematically the area can also be found by the following-
Area A can be found by the formula- [pic]
Area B can be found by the formula- [pic]
Total Area = 1 unit2
Area B = [pic]
= [pic]
Area A = [pic]
= [pic]
Ratio of area A : area B
[pic] : [pic]
= 2 : 1(satisfies the conjecture n:1)
Now I will investigate the ratios of area A and B with other natural numbers.
This means that the function will stay y=xn but here only n will change. The parameters will stay the same a=0 and b=1.
The ratios when n =3
The graph of y=x3
[pic]
Y=xn where n=3.
The Area of the shaded region or area B was found to be 0.25 units2.
Area A = 1- Area B = 1- 0.25 = 0.75 units2
Mathematically the area can also be found by the following-
Total area = 1unit2
Area B = [pic]
= [pic]
Area A = [pic]
= [pic]
Ratio of area A : area B
[pic] : [pic]
= 3 : 1(satisfies the conjecture n:1)
Now I will investigate the ratio of the areas for n being a rational number.
The ratio when n= [pic]
Y = x0.5
[pic]