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IB Math Portfolio Investigating Ratios

Extracts from this document...

Introduction

Investigating Ratios of Areas and Volumes

The aim of this portfolio is to investigate the ratio of areas form when  is graphed between two arbitrary parameters  and  such that .

1. Given the function, consider the 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.

Calculate the ratio of the areas for other functions of the type ,  between and .

Let ,

Finding the ratio of area A: area B:

The ratio of area A: area B is 3:1.

Let ,

Finding the ratio of area A: area B:

The ratio of area A: area B is 4:1.

Let ,

Finding the ratio of area A: area B:

Middle

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.

1. 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:

Conclusion

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:

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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