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IB Math Portfolio Investigating Ratios

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Introduction

Investigating Ratios of Areas and Volumes

The aim of this portfolio is to investigate the ratio of areas form when image02.pngimage02.png is graphed between two arbitrary parameters image54.pngimage54.png and image55.pngimage55.png such that image56.pngimage56.png.

image117.jpgimage00.pngimage01.png

  1. Given the functionimage03.pngimage03.png, consider the region formed by this function from image11.pngimage11.png  to image23.pngimage23.png  and the x-axis. This area is labeled B. The region from image24.pngimage24.png  and image35.pngimage35.png and the y-axis is labeled A.

Finding the ratio of area A: area B:

image52.png
image57.png
image60.png

image62.png
image65.png
image67.png

image07.pngimage07.png The ratio of area A: area B is 2:1.

image77.pngimage82.png

Calculate the ratio of the areas for other functions of the type image02.pngimage02.png, image95.pngimage95.png between image19.pngimage19.pngand image20.pngimage20.png. image88.png

Let image96.pngimage96.png,

image42.pngimage42.png

image45.pngimage45.png

Finding the ratio of area A: area B:image97.png

image98.png
image99.png
image100.png

image101.png
image102.png
image103.png

image07.pngimage07.png The ratio of area A: area B is 3:1.

Let image104.pngimage104.png,

image66.pngimage66.png

image105.pngimage105.png

Finding the ratio of area A: area B:

image106.pngimage107.png

image07.pngimage07.png The ratio of area A: area B is 4:1.

Let image108.pngimage108.png,

image109.pngimage109.png

image110.pngimage110.png

Finding the ratio of area A: area B:

image111.pngimage112.png

image07.pngimage07.png

...read more.

Middle

image129.png
image130.png

image04.png
image05.png
image06.png

image07.pngimage07.png The conjecture is not true to negative integers.

Irrational Numbers:

Let image08.pngimage08.png,

image09.pngimage09.png

image10.pngimage10.png

Finding the ratio of area A: area B:

image12.png
image13.png
image14.png

image15.png
image16.png
image17.png

image07.pngimage07.png The ratio of area A: area B is image18.pngimage18.png:1. This means the conjecture is true to irrational values of n.

  1. This conjecture is further tested for areas not only limited between image19.pngimage19.pngand image20.pngimage20.png.

image21.pngimage21.png  , for region formed by this function from image11.pngimage11.png  to image22.pngimage22.png  and the x-axis. This area is labeled B. The region from image24.pngimage24.png  and image25.pngimage25.png and the y-axis is labeled A.

image26.pngimage26.png

Finding the ratio of area A: area B:

image27.png
image28.png
image29.png

image30.png
image31.png
image32.png

image33.pngimage34.png

image07.pngimage07.png The ratio of area A: area B is 2:1.

image21.pngimage21.png  , for region formed by this function from image23.pngimage23.png  to image22.pngimage22.png  and the x-axis. This area is labeled B. The region from image35.pngimage35.png  and image25.pngimage25.png and the y-axis is labeled A.

image26.pngimage26.png

Finding the ratio of area A: area B:

image36.png
image37.png
image38.png

image39.png
image40.png
image41.png

image07.pngimage07.png

...read more.

Conclusion

Then take into account the function image66.pngimage66.png, and the region formed by this function from image23.pngimage23.png  to image22.pngimage22.png  and the x-axis (Area A). To find the area formed on the y-axis (Area B), substitute image54.pngimage54.png and image55.pngimage55.png values as 1 and 2, into the equations image68.pngimage68.png and image69.pngimage69.png, making image35.pngimage35.png and image70.pngimage70.png.

Calculating the ratios of areas A:B:

image71.png
image72.png
image73.png

image74.png
image75.png
image76.png

image07.pngimage07.png The ratio of area A: area B is 4:1.

Next take into account the function image42.pngimage42.png, and the region formed by this function from image23.pngimage23.png  to image22.pngimage22.png  and the x-axis (Area A). To find the area formed on the y-axis (Area B), substitute image54.pngimage54.png and image55.pngimage55.png values as 1 and 2, into the equations image78.pngimage78.png and image79.pngimage79.png, making image35.pngimage35.png and image80.pngimage80.png.

Calculating the ratios of areas A:B:

image81.png
image83.png
image84.png

image85.png
image86.png
image87.png

image07.pngimage07.png 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 image02.pngimage02.png from image54.pngimage54.png and image55.pngimage55.png and for the regions stated.

To further support the conjecture:image89.png

image90.png

image91.png


image93.pngimage92.pngimage94.png

...read more.

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

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