Question

Mark is building a pen for his dog and he got 24 feet of free fencing from Lowes. How many ways can he build the pen and which one is the best for his dog. Explain your reasoning.

Answer 1

Mark build a pen such that the parimeter of the pen is 24 Let x & y are length and width of the pen then parimeter= 2(x+y)= 24 which implies x+y=12. No . of ways he build a pen is no . of ways he choose x,y such that x+y=12. The pen is best for dog if the area of pen is maximum . So maximize the area A=xy such that x+y=12 which implies A=x(12-x)=12x-x² then. dA/dx=12-2x=0 implies x=6 and d^{​​​​​​2}​​​​​A/dx^{​​​​​2}=-2<0 at x=2 . Therefore Area is maximum at x=y=6 by Maxima and minima of functions of one variable. Hence if we choose length and width of pen is 6 then it is best for dog