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Want to get useful Information with the given functions

GROUPS:
Hi, I'm a student of civil engineering, Regardless of that, I need to do something regarding this functions, even though they are fairly easy to understand, my main objective is to analyze each set of equations with their given values ranges, to get any useful information about them, to determine what are equations that by a given X (Price), can lower the values given for the Y (Quantity). The constrains on analizing the data is that I want to know what ranges of prices is better to sell  on each Category, given a limited quantity, on integers as from 1 up to 350. as Gold Premium, Gold and Silver Categories. More Specifically, by a determined quantity to sell, I want to know what range of inputs can get the lower range of outputs for overall categories. For example I want to sell 20 Tvs and each cost 350$ and I want to know what range of prices for that TV to be sold gives me the lowest inputs (That are the cost for selling them Online). These Functions are charts for selling them. I want to optimize my costs for selling them online.

 X = Price
Y = Quantity


                  Gold Premium
Functions                                  Values

0< x*y < 8000                             400
7999.99<x*y< 25000              0.05*x*y
24999.99 <x*y                           1250

                   Gold
Functions                                  Values
0 < x*y <3000                              90
 2999.99 < x*y < 24000           0.03*y*x
23999.99 < x*y                             720

                      Silver
Functions                             Values
0<x*y<600                                 6
599.99<x*y<37000             0.01*x*y
36999.99<x*y                         370


By just looking I can Visualize the loop holes but I'm for some reason, don't know where to start. And I want the Big Picture how to solve this, or how to get the data in a chart.

I appreciate if I can get the answer, as this isn't for homework or similar, is for my instructional purposes, as I want to know how to start thinking all by myself, but I need practice, I just visualize the problem, and what I want, now I Just need to know how to start using the mathematical tools I need.

Thanks for any help.
POSTED BY: Eduardo Pinto
Answer
8 months ago
Starting with Piecewise functions in terms of v = x*y
 goldPremiumFunction = Piecewise[{
    {400, 0 < v < 8000},
    {0.05*v, 7999.99 < v < 25000},
    {125, 24999.99 < v}}]
 
 goldFunction = Piecewise[{
    {90, 0 < v < 3000},
    {0.03*v, 2999.99 < v < 24000},
    {720, 23999.99 < v}}]

silverFunction = Piecewise[{
   {6, 0 < v < 600},
   {0.01*v, 599.99 < v < 37000},
   {370, 36999.99 < v}}]
plot v up to 40,000
Plot[{goldPremiumFunction, goldFunction, silverFunction},
{v, 0, 40000}]
then inject x*y to replace v to show the x, y surfaces.
Plot3D[
   With[{v = x*y}, {goldPremiumFunction, goldFunction, silverFunction}],
{x, 0, 200}, {y, 0, 200}]
POSTED BY: Christopher French
Answer
7 months ago