Constructing an Ornamental Pool

[Larry Moats]

How would someone go about constructing a hypothetical pool, using functions to generate the shape of that pool on a graph. The functions and associated shape must meet the following requirements

a. Surface area approximately 35m² (± 10%)
b. Average depth 1m. i.e. volume 35m³ (± 10%)
c. The shape of the pool should consist of a curve or curves, (not circular)

d. The cross-section of the pool across its longest dimension should be a curve of some complexity

e. all curves should be represented by functions.

f. proof of reasoning should be presented

 

[stapel]

How would someone go about constructing a hypothetical pool, using functions to generate the shape of that pool on a graph. The functions and associated shape must meet the following requirements

a. Surface area approximately 35m² (± 10%)
b. Average depth 1m. i.e. volume 35m³ (± 10%)
c. The shape of the pool should consist of a curve or curves, (not circular)

d. The cross-section of the pool across its longest dimension should be a curve of some complexity

e. all curves should be represented by functions.

f. proof of reasoning should be presented

"How would someone go about" this task? By experimenting with various functions, and seeing what happens. Then making adjustments to get closer to the desired result.

In other words, there are very many "correct" answers to this exercise. Your particular answer will depend upon your particular efforts. So please tell us: What are your thoughts? What have you tried so far? Where are you getting stuck?

Please be complete. Thank you!

 

[Larry Moats]

Specific Function

Alright so I have the parametric function x(t)=((5-1) cos(t)+1 cos(t(5-1)/1),

y(t)=(5-1) sin(t)-1 sin(t(5-1)/1)

What I need to prove is the aforementioned requirements in the previous post for this function, firstly the surface area

To do this it is recommended that I do so using the Monte Carlo Method, The Trapezoidal Method and Integration by Rule, but am unsure of how to proceed.

I have attached the shape generated for ease of reference



Much Appreciated,

Larry

 

[stapel]

Alright so I have the parametric function x(t)=((5-1) cos(t)+1 cos(t(5-1)/1),

y(t)=(5-1) sin(t)-1 sin(t(5-1)/1)

How did you obtain these functions? What is the domain of t? Which of x and y is the shape of the pool? Which is the depth function?

Please be complete. Thank you!