A rancher wants to fence in an area of 3000000 square feet in a rectangular field and then divide it in half with a fence down the middle parallel to one side. What is the shortest length of fence that the rancher can use?
2007-11-09 14:18:31 · 2 answers · asked by Rachel 1 in Science & Mathematics ➔ Mathematics
Let the area rectangle be xy
x = length, y = width,
a dividing fence built across is y.
The perimeter p = 2x + 3y
Area = 3 million
3 = xy ===> x = 3/y
Substitute into the perimeter eqn.
Unit is in million.
p = 2(3/y) + 3 y
p = 6y^-1 + 3y
Optimize by differtiation
p'= -6y^-2 + 3
0 = -6y^-2 + 3
6y^-2 = 3
y = sq.rt 2
Solve x, x = 3/sqrt 2
Substitute x and y into the perimeter.
p=2(3/sqrt 2) + 3 (sqrt 2)
p= 6/sqrt + 3 sqrt 2
p=6 sqrt 2 million ft
p= about 8.485 million ft.
2007-11-09 17:07:03 · answer #1 · answered by mlam18 6 · 0⤊ 0⤋
So, what's the shortest length a side of a rectangle with area of 3X10^6 can have? That allows you to work it out.
2007-11-09 22:24:14 · answer #2 · answered by John R 7 · 0⤊ 2⤋