Water Bottle Design Using Polynomials

[BigKevSebas]

Homework Statement

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I am to design a 600mL water bottle by drawing one side (bottle lying horizontally). Three types of functions must be included (different orders). The cross-sectional view would be centred about the x-axis, and the y-axis would represent the radius of that particular section. There are meant to be no gaps and a "smooth transition between curves must be present.

Homework Equations

All they have given us is V=πr^2
and V=π∫y^2 dx.

The Attempt at a Solution

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I don't know even know where to start.

 

[scottdave]

You are going to need to do some trial and error with some polynomials. At the transition points, the function values and their derivatives must match to make the smoth transition. The volume integral should wind up equal 600. Remember that mL is the same as cm^3

 

[LCKurtz]

Homework Statement

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I am to design a 600mL water bottle by drawing one side (bottle lying horizontally). Three types of functions must be included (different orders)..

Different orders? Are you implying you must use only polynomials? Are you restricted to exactly $3$ functions or can you use more?

 

[LCKurtz]

Here's another suggestion, assuming you aren't stuck with polynomials. Even if you are, it might give you some idea how to approach the problem. Consider the figure below:

Say each square is of side length $h$. You can scale and translate a cosine curve to make the upper left smooth curve and a quarter circle to make the upper right curve. Once you have the equations of the upper half, calculate the volume of rotation. Then choose $h$ to make it come out $600$. I'm thinking something like Maple would be very helpful. Good luck.

 

[Nidum]

@LCKurtz Similar idea to yours .

Having one horizontal line segment makes the rest of the construction relatively easy .

I don't think there is any problem scaling the functions to get the correct volume .

 

[LCKurtz]

Heh heh. But I want my jug to be able to sit on the counter and hold my beer.

 

[Nidum]

Here you are - a flat bottom jug for you .

 

[scottdave]

View attachment 208250

Here you are - a flat bottom jug for you .

I guess one could argue that the vertical line is not really need to be one of the 3 (there is no dx to integrate). As long as the derivative at that point matches a vertical line... And did they mean you must use exactly 3 functions, or at least 3?

 

[scottdave]

Here's another suggestion, assuming you aren't stuck with polynomials. Even if you are, it might give you some idea how to approach the problem. Consider the figure below:
View attachment 208241
Say each square is of side length $h$. You can scale and translate a cosine curve to make the upper left smooth curve and... Then choose $h$ to make it come out $600$. I'm thinking something like Maple would be very helpful. Good luck.

Fun fact: So this is similar to the way they create font characters (without the rotation to calculate volume). Each character is defined by some functions, but related to some scale factor, such that it can be scaled to any size from microscopic to billboard size, and still look the same, without any jagged edges. I searched for a reference link to add in, but every search I tried doing came up with fonts that represent mathematical symbols.

 

[LCKurtz]

Probably the most natural way to do such problems is to use cubic splines. Lots of info on the internet if you're interested. Just Google it if you want more info. One link, as an example, is:
http://www.math.ucla.edu/~baker/149.1.02w/handouts/dd_splines.pdf

 

[scottdave]

Probably the most natural way to do such problems is to use cubic splines. Lots of info on the internet if you're interested. Just Google it if you want more info. One link, as an example, is:
http://www.math.ucla.edu/~baker/149.1.02w/handouts/dd_splines.pdf

Thanks for the cubic splines article. That was interesting.

 

[Kaura]

Well I think Desmos Graphing Calculator would be your friend in this case. For the smooth transitions just make sure that the derivatives of the two interchanging functions are the same. You could just transition at the extrema and or horizontal points of the different functions. As for the 600ml part I guess you could come up with the functions first and then cut off the last function when the total volume intergral equals 600ml or whatever. I am interested to see what you come up with and I think I will try this problem myself.

 

[BigKevSebas]

Thankyou for the help. Helped me complete the task.