Exercises 4.3.8 Page 177.

  1. A rancher has 1000 feet of fencing in which to construct adjacent, equally sized rectangular pens. What dimensions should these pens have to maximize the enclosed area?

(Page 177).

Let’s say width = x, and lenght =y 3x + 2y =100

y = (1000 - 3x)/2

Area = (1000x - 3x\(^2\))/2 = 500x - (3/2)x\(^2\)

Area’ = 500 -3X (point where area is maximum)

fx <- function(x) {
  500 - (3*x)
}

a <- uniroot(fx, c(200, -200))
a$root
## [1] 166.6667
x = a$root
x
## [1] 166.6667
y = (1000 - 3*x)/2
y
## [1] 250
Max_Area = x*y
Max_Area
## [1] 41666.67

So the maxium of enclosed area is 41666.67 square feet