A farmer is setting up separate pens for livestock by fencing off a rectangular area and running fence to create three parallel pens. If the farmer has to enclose a total area of 9,800 square meters, what is the shortest length of fence needed

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

22 dimes and 9 nickels

step-by-step explanation:

the answer would be d.

step-by-step explanation:

move all variables to the left side and all constants to the right side

(x−3)^2+(y−1)^2=16

this is the form of a circle. use this form to determine the center and radius of the circle.

( x − h ) 2 + ( y − k ) 2 = r 2

match the values in this circle to those of the standard form. the variable   r   represents the radius of the circle,   h   represents the x-offset from the origin, and   k   represents the y-offset from origin.

r = 4

h = 3

k = 1

the center of the circle is found at   ( h , k ) .

center:   ( 3 , 1 )

these values represent the important values for graphing and analyzing a circle.

center:   ( 3 , 1 ).

radius:   4