Introduction to Modern Mathematical Modeling with R:

A User’s Manual to Train Mathematical Consultants

 

A Cambridge University Press book by

SSP Shen

 

 

R Code written by Dr. Samuel Shen, Distinguished Professor
San Diego State University, California, USA
https://shen.sdsu.edu
Email:

 

Compiled and Edited by Joaquin Stawsky
San Diego State University, August 2022

 

 

 

Chapter 6: Mathematical Modeling By Calculus

Optimal Dimensions of Food Cans

pfr <- function(r) {6*pi*r^3 + 9.4*pi*r^2 -1456.88} 
uniroot(pfr,c(0,8))
## $root
## [1] 3.796289
## 
## $f.root
## [1] -0.0001021878
## 
## $iter
## [1] 8
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 6.103516e-05

 

Optimal Production Level of Oil

x <- seq(7.5,11.0,length=100)
rev <- function(x) {10^(-3)*30*x*(10000/(x+1)^2)*(cos(0.5*(x-8.5)))} 
plot(x,rev(x),type="l", 
     main="Monthly Oil Revenue vs. Oil Production for a Country",
     xlab="Oil Production [Mbbl/Day]",ylab="Billion USD" )
grid(nx = NULL, ny = NULL)

#Search for the maximum revenue
max(rev(x))
## [1] 28.75984
#Search for the optimal production level for the maximum revenue op=0
for(k in 1:100){
  if(rev(x)[k] > 28.7598){
    op=x[k]} 
  }
op
## [1] 8.136364
crrev <- function(x) {(10000/(x+1)^2)*(cos(0.5*(x-8.5)))- 
    0.5*x*(10000/(x+1)^2)*(sin(0.5*(x-8.5)))- 
    2*x*(10000/(x+1)^3)*(cos(0.5*(x-8.5))) } 
uniroot(crrev,c(7.5,11))
## $root
## [1] 8.120058
## 
## $f.root
## [1] -0.0001251906
## 
## $iter
## [1] 5
## 
## $init.it
## [1] NA
## 
## $estim.prec
## [1] 6.103516e-05
x <- seq(7.5,11.0,length=100)
mrev <- function(x) {10^(-3)*30*x*exp(0.5*(x-10.0))*(10000/(x+1)^2)*(
  cos(0.5*(x-8.5)))}
plot(x,mrev(x),type="l", main="Market-Modified Oil Revenue vs. Oil
     Production for a Country", xlab="Oil Production [Mbbl/Day]",
     ylab="Billion USD" )
grid(nx = NULL, ny = NULL)

#Search for the maximum revenue
max(mrev(x))
## [1] 18.18665
#Search for the optimal production level for the maximum revenue op=0
for(k in 1:100){
  if(mrev(x)[k] > 18.18664){
    op=x[k]} 
  }
op
## [1] 9.90404