Geometry Question Part B?

Question

Ok here is part 2 of 2.

Keli and Mario are planning to plabt rectangular gardens of the same length, side by side with fencing all around and dividing the two plots. The total amount of fencing is 100 feet. If the total area of the two plots is 336 ft. squared and the dimensions are integers, what is the length of the fence that divides the two plots?

Here is a diagram for reference. http://i58.photobucket.com/albums/g254/poker5495/POWVI.jpg

10 points will go to correct and informative answer as well as 1 thumbs up from me.

Answer 1

Area is 336 sq.ft and you have 100 feet of fencing

Dimensions will be 24 by 14
Fence around both the gardens will be
24 + 24 + 14 + 14 = 76
The fence dividing the 2 gardens is the same length as the two outside fences which is 24 feet.
76 + 24 = 100

The length of the fence that divides the two plots is 24 feet.

Answer 2

Let L be the length of the two plots and W be the combined width of the two plots. The total amount of fencing required is 2L+2W for the perimeter plus L for the fence dividing the two plots, so we have 100=3L+2W. Since the total enclosed area is 336, we have that LW=336 or W=336/L. Substituting this into the first equation:

100=3L+2*336/L

Multiplying by L:

100L = 3L² + 672

Subtracting 100L:

3L² - 100L + 672 = 0

Employing the quadratic formula:

L=(100±√(100² - 4*3*672))/6
L=(100±√1936)/6
L=(100±44)/6

L=24 or L=28/3

However, since we are told the dimensions are integers, we may disregard the second solution. So the length of the dividing fence is 24 ft (also, simple division yields that the combined width of the two plots is 14 ft, but that's not important right now).

Answer 3

Say that the short side of the fence is 2x and the long side of the fence is y. The perimeter equation for the fence is:

x + x + x + x + y + y + y = 100
4x + 3y = 100

The area equation is:

2xy = 336
xy = 168

You can solve two simultaneous equations. From the area equation, x = 168/y. Substituting this value for x into the perimeter equation gives:

4(168/y) + 3y= 100

672/y + 3y = 100 (multiply both sides by y)

672 + 3y^2 = 100y

3y^2 - 100y + 672 = 0 (divide by 2)

The only integer solution to this y = 24, so the length of the fence is 24 ft.

Answer 4

ok...

we've got ourselves a big rectangle and a line in the middle of it (dividing the 2)

the amount of fencing of this rectangle is L + W + L + W + W
we add Width another time, because the line in the middle of the rectangle, so we've got an equation:
2L + 3W = 100
The area of this rectangle = 336... so we have another equation:
LW = 336

now we just solve by substitution
L = (100 - 3w) / 2 <-- solve first equation for L
(100 - 3w)/2 * W = 336 <--substitute
50w - 3w^2/2 = 336
-3w^2/2 + 50w - 336 = 0
using quadratic formula we get the w = 9.33333333 or 24....

Both of these solutions work.