Answer :
Answer:
L(min) = 560 m
Step-by-step explanation:
Dimensions of rectangular area are: "x" and "y"
Then x*y = 9800 m²
If this area have to be divided into three parallel pens total length of fencing material would be:
L = 2*x + 4*y
And L as a function of x is
x*y = 9800 ⇒ y = 9800/x
L(x) = 2*x + 4 * 9800/x ⇒ L(x) = 2*x + 39200/x
Taking derivatives on both sides of the equation we get
L´(x) = 2 - 39200/x²
L´(x) = 0 ⇒ 2 - 39200/x² = 0
2*x² - 39200 = 0
x² = 39200 / 2 ⇒ x² = 19600
x = 140 m
And y = 39200/ 140
y = 70 m
Then minimum length of fence is:
L(min) = 2*140 + 4*70
L(min) = 280 + 280
L(min) = 560 m
Checking in the second derivative:
L´´(x) = 2*39200/x³ > 0 then we get a minimum of the function L at the point x = 140