Answer to Question #342410 in Calculus for yousii

Question

A farmer wants to fence an area of 60,000 m² in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be (in m) in order to minimize the cost of the fence?

smaller value = m

larger value = m

Answer 1

The total length will be x

and the height will be y

Needed Equations:

Perimeter of this diagram

P=2x+3y

Total Area A=xy=60000

Solve for y

y=60000/x

Substitute the equation for y into the function for perimeter.

P=2x+3(60000/x)=2x+180000/x

Find the derivative of the equation for perimeter

P'=2-180000/x²

Use the derivative equation in order to find the critical point(s) that minimize the perimeter.

Critical points are when

P'=0 and when P' does not exist. It is also good to check the endpoints of an equation in order to check for a maximum or minimum.Since P' always exists, only find where P'=0

(there will be no endpoints to check since this is a field).

0=2-180000/x²

$x=\sqrt{180000/2}=300$

P' is positive when x>300 and P' is negative when x<300 , therefore meaning that x=300 is a minimum. Since this value is a minimum, the perimeter (and the cost) is minimized when the total length is 300 m.

Find the height ( y ) when x=300

y=60000/300=200

smaller value =200 m

larger value =300 m