Hippo takes a plunge

An in-class activity summarized and analyzed:

Our task was to calculate how many rubber bands were needed to allow a plastic hippo to bungee jump the high of a flight of stairs. The goal was to minimize error and optimize the distance of the fall without exceeding it (ie. not cashing to the floor).

We first attached one rubber band to the foot of the hippo and tested its maximum fall distance. We then added a second rubber band and did the same. We repeated this for the first five rubber bands and then the seventh. We noted that each rubber band averaged an additional 22 cm. to the fall distance. I, myself, speculated that with each rubber band added, the additional distance to the fall was slowly decreasing.

Since the height of the jumping platform to the ground was about 505.46 cm. (199 in.) we calculated that 22 rubber bands would suffice. We found that with seven rubber bands, the hippo dropped a total of 156.5 cm. By calculations, we determined that with 21 rubber bands, (156.5(3)) the hippo would drop a total of 469.5 cm. With an additional 22 cm an extra rubber band would allow - increasing our rubber band count to 22 - we speculated that the hippo would drop a total of 491.5 cm.. This leaves the hippo just 13.96 cm. off of the ground. We determined that with averaging the cm/rubber band, 13.96 cm. from the ground leaves room for unaccounted error.

Below is a recorded account of the bungee jump:

Above is an account of our second jump. It should be added that the first jump was roughly 40 cm. away from the floor however when dropped, the rubber bands bunched up and did not allow for a clean drop. Thus, we dropped again. On the second drop, we ended up a mere17 cm. from the floor. This leads to an error of 3.04 cm. away from the floor based on our calculations.

 This activity made me think of limits. When it came to the distance each rubber band added allowed the hippo to drop, the number, ultimately, slowly shrank. However I cannot imagine that if one were to add an obscenely large amount of rubber bands (x rubber bands tending towards infinity) that the last rubber band added would allow a distance of 0 cm more distance dropped. Thus, the limit of what each rubber band would contribute to the drop would have to be greater than zero, but less than the last distance recorded (being 18cm). I should add that I do not know this number and that more testing would be involved to calculate it but this is just a speculation.

Also, it had me wondering about elasticity. According to Anthony Coy, "When the rubber bands were heated, the particles stretched out, making them more elastic and able to withstand greater force" (2005). This has me curious, wondering what would happen to the limit a rubber band might add to the drop when they are heated: would the limit be greater than that of the room-temperature bands or would the limits be equal?

Below is an account for our rough data:

Al Kharaji Summation

Given the summation

find an equation that defines it.
I start by constructing the sequence xn.

 That is,

Using exploration and the series, I derived an equation that defined the above summation:

 See scratch work below:

First I tried different values of n. That resulted in the circled numbers: 1, 9, 36, 100, and 225. I noticed that all these numbers were perfect squares with the corresponding numbers 1, 3, 6, 10, 15, 21 as their square roots. These numbers form a sequence defined as xn above. I used this sequence to derive a way to find the summation of any n.

Tessellations

Above is the tessellation I completed in class. I will continue to expand on this project.
From the looks of it, there are two complete figures above:

Figure 1: Yellow Hexagon               
               Red Trapezoids
                Orange Squares
               

Figure 2: Yellow Hexagon
                Blue Parallelograms
                Green Equal-lateral Triangles
                Tan Elongated Parallelograms

Note: Though the figures can be determined in multiple different ways because of connecting pieces, to make things easier I have divided things as such.

Reasons for the tessellation being incomplete is because of time constraints and limited material. If I had enough time and pieces, however, I would construct three more of Figure 1 and attach them to the bottom of Figure 2. This would complete the circle of Figure 1's around Figure 2. From there, I would construct multiple Figure 2's and fill in such spaces around the six Figure 1's. This process could continue on and on without spaces between Figures.

I researched other tessellations and included them below:

                                                  
 Credit                                                 Credit                                        Credit                     

The Nature of Mathematics

What is an axiom? 

 According to the Merriam-Webster online dictionary, an axiom is,

In mathematics terms, an axiom is typically a proposition or rule that does not require a proof for supporting evidence of its truth. Axioms can take different forms and can also be used in other areas of study other than mathematics such as law, the sciences, etc..

What is the role of axioms in mathematics?

Axioms in mathematics play a large role since mathematics focuses on proving conjectures. These conjectures are proved in a similar way of building blocks: you must lay a foundation and build upon those blocks. These foundations typically contain axioms. When considering axioms as building blocks, these do not need to be proven every use but simply recalled when used. It reminds me of the idea of infinity and how if we had to prove each axiom with its use would also then call for a proof of the axiom's axiom, and this could go on for an unfortunate long period of time. This causes too much unnecessary work. Thus, axioms are very convenient for mathematicians.

What is Math?

What exactly is mathematics, you ask? If you were to ask me that question even two years ago, my answer would be extremely different. At that time, I was still completing the Calculus sequences (Calc 1, Calc 2, and Calc 3). At that time, math meant strict calculations: whether it be addition, deriving, or integrating. At that moment in my life, math was very literal and very black and white.

My answer now includes those original thoughts, but goes much, much deeper. Now, I believe Mathematics to be a subject that encompasses everything logical. Mathematics is an tool to problem solve, whether that problem be dealt with numbers or patterns excluding numbers (such that Discrete has taught me). It is a subject that can account for so much of our world and our existence. Mathematics is a tool that, if used correctly, can explain many natural and man-made occurrences that surround us.