You’re building five identical pens adjacent to each other with a total area of 1000 m^2, as shown below. What dimensions should you use to minimize the amount of fencing needed?

Question

Answer 1

Steps for solving the problem

1) Find the function for fencing: F=10x+6y

2) Find the equation in terms of x and y: Area=1000=5xy, y=200/x

3) Rewrite F in terms of only one variable x: Sice y=200/x, F=10x+6y=10x+6(200/x)=10x+1200/x

F(x)=10x+1200/x

4) Find the minimum value of F(x) =10x+1200/x when 0<x<100 using the first derivative test

F'(x)=10-1200/x²=0, so critical numbers are ±2√3, 0, but only x=2√30 is between 0 and 100.

F' changes the sign from - to + at x=2√30, so F(2√30) is the minimum

5) Find the dimension to minimize the fencing.

x = 2√30 and y=200/x=200/2√3=100/√3=100√3/3

Answer 2

F = 10x + 6y

5xy=1000

xy=200

x=200/y

F = 2000/y + 6y

dF/dx = -2000/y² + 6 = 0 ....the solution will give you the value for x at minimum