Business Calculus       Advanced Application
               
Lagrange function
　This is an example of analysis using the Lagrange function and Excel solver when there cognition of a company can be  expressed  as a formula. If you're not good at integration, see Excel Solver.

*Market Environment
 According to a survey by a market research firm in the beverage industry, the company's awareness (target age: 15  years  old and over, under 25 years old: L;) is proportional to the cube of the sales volume of coffee (x) and the product of  the sales volume of  tea (y).
  L= x^3y.
Due to production facility limitations, the maximum sales volume of Z company's coffee and tea for the coming season in  this industry is as follows:
    x + 2y =16 billion yen.


*Question
 What is the combination of coffee (x) and tea (y) sales volumes that maximizes the company's awareness (L)?

*Answer
 Using the Lagrange function and the first-order condition...
                                     
By solving the above system of three equations,

the values of coffee (x)= 12 billion yen , tea (y) = 2 billion yen, L=3456, which is the maximum.
     
 
Alternative solution: Example solution using Excel's solver  
                                              
   
 We got the same results of above : coffee (x) = 12 billion yen ,tea (y) = 2 billion yen, L=3456, which is the maximum.
 

           A quick note on business calculus :  The area of a figure formed by a string of length L