Posted by **Jackie** on Sunday, February 10, 2013 at 2:30am.

A farmer uses 1034 meters of fencing to enclose a rectangular region and also to subdivide the region into three smaller rectangular regions by placing a fence parallel to one of the sides. Find the demensions that produce the greatest enclosed area.

- Math -
**Reiny**, Sunday, February 10, 2013 at 8:00am
After making your sketch and creating the 3 rectangles

let the length of each rectangle be x, let the width of each rectangle be y

So the area of the whole enclosed area = 3xy

also : 4x + 6y = 1034

6y = 1034 - 4x

y = (1034 - 4x)/6

area = 3xy - 3x(1034-4x)/6

= 517x - 2x^2

If you know Calculus ....

d(area)/dx = 517 - 4x

= 0 for a max of area

x = 517/4 =129.25

y = 86.167 m

(maximum area = 3xy = 33411.125

Check by taking a value of x slightly smaller and slightly larger and finding their areas

e.g

x =129 , then y = 86.3333

area = 3xy = 33411 , just a bit smaller

x = 130, then y = 85.6666..

area = 3xy = 33410 , again a bit smaller

my answer is plausible.

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