Question

A rancher wants to fence in a rectangular area of 24600 square feet in a field and then divide the region in half with a fence down the middle parallel to one side. What is the smallest length of fencing that will be required to do this?

Answer 1

120
`√(41)` ft of fencing

Explanation:

24600 = xy

Fence (F) = 3x + 2y

24600/y = x

F = 3(24600/y) + 2y

F = 73800/y + 2y

dF/dy = -73800/y^2 + 2

2 = -73800/y^2

y^2 = -73800/2 = 36900

y = 30
`√(41)`

24600 = x(30
`√(41)`)

24600 / 30
`√(41)` = x = 20
`√(41)`

total fencing = 3(20
`√(41)`) + 2(30
`√(41)`) =
`√(41)`(60+60) = 120
`√(41)`

Answer 2

784.2 ft

Explanation:

Area= xy

Smallest paramenter and so smallest fencing will occur when x=y

Area= x²

24,600= x²

x= 156.84

Length of fencing= 5x

= 5(156.84)

= 784.2 ft