Finding the ratio of area A: area B:
The ratio of area A: area B is 10:1.
From the findings above, it is evident that when , the ratio of area A: area B is always n : 1 when n is a positive integer more than 0.
To further investigate my conjecture, test the conjecture for other subsets of the real numbers:
Fractions:
Let ,
Finding the ratio of area A: area B:
The ratio of area A: area B is :1, which means that the conjecture is true to fraction values of n.
Negative Integers:
Let ,
Finding the ratio of area A: area B:
The conjecture is not true to negative integers.
Irrational Numbers:
Let ,
Finding the ratio of area A: area B:
The ratio of area A: area B is :1. This means the conjecture is true to irrational values of n.
-
This conjecture is further tested for areas not only limited between and .
, for region formed by this function from to and the x-axis. This area is labeled B. The region from and and the y-axis is labeled A.
Finding the ratio of area A: area B:
The ratio of area A: area B is 2:1.
, for region formed by this function from to and the x-axis. This area is labeled B. The region from and and the y-axis is labeled A.
Finding the ratio of area A: area B:
The ratio of area A: area B is 2:1.
, for region formed by this function from to and the x-axis. This area is labeled B. The region from and and the y-axis is labeled A.
Finding the ratio of area A: area B:
The ratio of area A: area B is 3:1.
From the above findings, it can therefore be concluded that my conjecture is not only limited to areas between and , but holds true to other values of as well - from and such that .
-
Considering if my conjecture is true for the general case from and and for the regions:
Area A: , , and the y-axis
Area B: , , and the x-axis.
In order to determine this, trial and error substitution is used.
Take into account the function , and the region formed by this function from to and the x-axis (Area A). To find the area formed on the y-axis (Area B), substitute and values as 0 and 1, into the equations and , making and . The ratio formed between area A and area B is as calculated above in Q1, 2:1.
Then take into account the function , and the region formed by this function from to and the x-axis (Area A). To find the area formed on the y-axis (Area B), substitute and values as 1 and 2, into the equations and , making and .
Calculating the ratios of areas A:B:
The ratio of area A: area B is 4:1.
Next take into account the function , and the region formed by this function from to and the x-axis (Area A). To find the area formed on the y-axis (Area B), substitute and values as 1 and 2, into the equations and , making and .
Calculating the ratios of areas A:B:
The ratio of area A: area B is 3:1.
From the above three trials we can successfully conclude that my conjecture is true for the general case from and and for the regions stated.
To further support the conjecture:
