The topic of this investigation is differential calculus and its
applications. The main purpose of this investigation is to further the
knowledge of differential calculus by finding the maximum area of a triangle
formed on an A4 paper via optimization and other different methods, as well as
researching and investigating a general rule for solving this type of problem.
The investigation is going to be six sections. During the development of
this investigation, technology and electronic devices will be used to assist and
improve the investigation. For example, calculator will be used to perform
complicated calculations, and Grapher will be used to draw diagrams. In
addition, all diagrams in this investigation are drawn with scale.
First, a reasonable conjecture of the position of A will be formed
so that S∆ACX (“S” represents the area) reaches
maximum. Second, this conjecture will be proved or disproved in the following
section, and updates or further refinements will be applied if it is disproved.
In the third section, calculus will be introduced to find the exact
position of A and the maximum area of S∆ACX. After
this, two alternative methods will be applied to find the answer.
The fifth section is going to be a research and investigating part, where
different sizes of the paper series will be investigated. This section is
aiming at finding if there is a similar pattern or general rule that could be
applied to all sizes of paper series. There will be a conjecture for each paper
size, these conjectures will then be tested for their accuracy. And at the end
these conjectures will be compared to see if a similar patter exists.
The last section is an extended section. This section is planned to
explore some of the findings or ideas appeared during the development of this
Finally, there will be a conclusion that summaries the main idea and key findings that discovered and demonstrated in this investigation. There will also be a discussion on the reasonableness of the results and the limitation of the methods.
Conjecture of the Position of A
The width of a piece of A4 paper is 210mm. Therefore, when folding
the paper, the furthermost position for A on CD is 210mm
away from C.
increases from 0mm, S∆ACX increases.
decreases from 210mm, S∆ACX increases as well.
∴ The maximum value of S∆ACX is likely to occur around half of the
maximum value of
The maximum value of S∆ACX possibly occurs at
Therefore, the conjecture is formed:
S∆ACX will reach its maximum when
Proof of the Conjecture
The approximate range that the maximum value of S∆ACX occurs
can be determined using trial and error method.
Divide AC into 21 equal segments, so that every segment is 10mm
in length. Fold the paper 21 times, each time let A be on a different graduation
and measure the length of CX, let CX = y. Then calculate the
approximate area of ∆ACX using
. The measurement and results are
presented in this table.
|x (mm)||y (mm)||S∆ACX (mm2)|
From the table, it is clear that S∆ACX reaches its maximum value between x = 110mm and x = 130mm.
As such, the conjecture is slightly inaccurate since there is an approximately 5mm
to 25mm difference. The refined conjecture can now be formed:
The maximum area of ∆ACX will occur between x = 110mm and x = 130mm.
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