Assignment Help! Maths Methods Water Slide

[sealtheseal]

Hello, I have recently gotten a Maths assignment for Year 11. I would really like to ace this paper but I'm currently a little stuck even after a large amount of research.

The assignment is about redesigning a safer version of the Verruckt Waterslide through the usage of Polynomial functions (only up to quartic) as well as piecewise functions. The height of the slide may remain the same or be shorter, however, the horizontal length must remain the same.We also have to include graphs and tables - and all i can think of is regression graphs and y and x variable tables for the verruckt slide and our new version.
 It also came to me that this also has aspects to physics in it, such as gravity, force and acceleration. How could I also tie this in?
We also have to assume many aspects of the waterslide. For example, the waterslide has not twists, all rafts are functional and safe etc.
Im not sure how to uniquely design the safer (new)version and how to justify how safe it would be through complex mathematical approaches. 

Here is the slide, some websites and what I have done so far on desmos:
https://www.desmos.com/calculator/t9cwmf1tlr
en.wikipedia.org › wiki › Verrückt_(water_slide)
http://www.independent.co.uk › News › World › Americas

If someone could please, give me a direction of how to take on and start this maths assignment, the formulas and what the new slide could look like, it would be really appeciated. im very lost Please guide me smart maths people!!

[K.Smithy]

Hey Sealtheseal!

I had a very similar assignment in Unit 1 QCE methods last year. We had to design three waterslides with piece-wise functions and then prove their reasonability.

I would recommend looking at international amusement park standards for waterslides (set by ASMT International). The guidelines put in place ensure that all slides:
- Don't exceed acceleration limits,
- Have appropriate emergency safety devices in place,
- Have suitable hydraulic systems,
- Are able to bear the load of passengers, water, etc.
You could make it an assumption that your new design abides by these standards.
(on the topic of observations and assumptions: don't forget to mention the scale of the graph (e.g. one unit on the graph = 1 meter of displacement), ensure that the graph doesn't contain any negative x-values (as a slide in real life can't have negative length) or y-values (unless your slide goes underground)).

Does your task sheet specifically state that you have to explore the physics behind water slides? If not, you can just state that as a limitation in your design (the fact that physics hasn't been taken into account).

As for designing a safer version and justify how it is safe, I would recommend just simply making the decline more gradual.
You could try graphing something like this:


But adding an extra drop (thats where you can bring in a polynomial function) so it more closely resembles the Verruckt Waterslide.

To justify how safe it would be all I can think of doing is making sure all the parts of you slide (the different functions) meet up. This can be achieved via the use of simultaneous equations. For example:

(the functions in this image are not the same ones in the last - but I used the same process for all three slides)
If you can show that the two functions meet up then you have proven the reasonableness of the design - if the points meet up, the slide is probably safe. I wouldn't want to go on a slide if all of the parts didn't meet up

Hope this somewhat helps, good luck!

[sealtheseal]

Thanks so much for the help! I really appreciate your time and effort in typing this up! Sorry, but I was just a little confused at when you were talking about how you worked out how the functions met up. If you had 3 functions used for your piece-wise could u still use the method that you put in? Does the point of insertion show this?
I will definitely use the info you gave me and rework it into my assignment! <3

[K.Smithy]

Thanks so much for the help! I really appreciate your time and effort in typing this up! Sorry, but I was just a little confused at when you were talking about how you worked out how the functions met up. If you had 3 functions used for your piece-wise could u still use the method that you put in? Does the point of insertion show this?
I will definitely use the info you gave me and rework it into my assignment! <3

No problem at all!
For three functions, you do the same thing. So if I used the slide in my last response as an example, what I would do is:
Show that y = 10 and y = -0.6x + 11.2 meet, through the use of a simultaneous equation:
10 = -0.6x + 11.2
0 = -0.6x + 11.2 - 10
0 = -0.6x + 1.2
-1.2 = -0.6x
2 = x
Then substitute that into one of the equations (for this example I have to substitute it into the second equation as the first doesn't have a variable)
y = -0.6(2) + 11.2
y = -1.2 + 11.2
y = 10
Point of intersect is (2, 10)
Then, do the same thing with y = -0.6x + 11.2 and y = 34/(x-3.5)
(For this one, I used technology because I didn't want to use my brain... but I ended up with x = 12)
Substitute in x = 12 into one of the equations (this time, any equation will work, as both have a variable - but I like to choose the easiest)
y = -0.6(12) + 11.2
y = -7.2 + 11.2
y = 4
Point of intersect is (12, 4)

So yeah, this process works the same no matter how many functions you have.
If you have 4 functions find the point of intersection between function 1 and function 2, then function 2 and function 3, then function 3 and function 4.
I hope this makes more sense