SOLUTION: Pls help me with this. It's really hard!! A farmer wants to make a rectangular garden of 7500 square meters, and she has 250 meters of fencing material. There is already a fence o

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Question 102929: Pls help me with this. It's really hard!!
A farmer wants to make a rectangular garden of 7500 square meters, and she has 250 meters of fencing material. There is already a fence on one side. What dimensions should her garden have?
I really need some explanation here. Pls help me. thanks.

Found 3 solutions by bucky, checkley75, Fombitz:
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website!
Let L represent the length of the rectangle and W represent the width of the rectangle.
.
The area of the rectangle is to be 7500 square meters.
.
The area of a rectangle equals the product of the length L and the width W or:
.
A = L * W
.
Now substitute 7500 for the area to change the equation to:
.
7500 = L * W
.
The 250 meters of fencing is a little tricky. This fencing will only be needed on 3 sides of
the rectangle because the 4th side is an existing fence. So we can say that the fencing material
will be used for the Length and the two Widths to be installed, and the existing fence will
be used for the other Length. So the 250 meters of fence will equal L plus 2W. In equation
form this is:
.
250 = L + 2W
.
Now return to the area equation. We can solve it for W by dividing both sides by L to get:
.
W = 7500/L
.
Substitute the right side of this for W in the fence length equation to get:
.
250 = L + 2W = L + 2(7500/L) = L + 15000/L
.
So start with
.
250 = L + 15000/L
.
You can get rid of the denominator by multiplying both sides of this equation (all terms)
by L to make the equation become:
.
250L = L^2 + 15000
.
To get this in "standard form" for solving, make the left side equal zero by subtracting
250L from both sides to change the equation to:
.
0 = L^2 - 250L + 15000
.
A little more standard form is obtained by simply transposing sides to get:
.
L^2 - 250L + 15000 = 0
.
The left side of this equation can be factored to:
.
(L - 150)(L - 100) = 0
.
This equation will be true if either of the factors equals zero ... because a multiplication
by zero on the left side will make the left side of the equation equal the zero on the
right side.
.
Setting each factor equal to zero means:
.
L - 150 = 0 or L = 150
.
and L - 100 = 0 or L = 100
.
Let's first choose L = 150 meters.
.
That leaves 100 meters of fencing material (250 meters - 150 meters) to use in making the
2 widths. So each of the widtha is 100 meters divided by 2 or 50 meters.
.
So this solution is to make the rectangle 150 meters by 50 meters which multiplies out
to give an area of 7500 square meters and use the 250 meters of fence as 2 widths of
50 meters each and a length of 150 meters. 150 meters of the existing fence is used for
the 4th side
.
How about our other potential solution of L = 100 meters? That means that if you cut 100 meters
of fence for the length, you have 150 meters left over for the 2 widths. So each width
is 75 meters. This makes the dimensions of the garden 100 by 75 meters. This means that
the area is 100 times 75 which is 7500 square meters. And the 100 meters plus 75 meters plus
75 meters for the sides adds up to the 250 meters of fencing. The existing fencing can
supply the missing 100 meter length.
.
So there are two answers to this problem. One is to make the rectangle 50 by 150 meters,
using the existing fence to provide one of the 150 meter sides. The other is to make
the rectangle 75 by 100 meters, using the existing fence to provide one of the 100 meter
sides.
.
Hope this helps you to understand the problem.
.

Answer by checkley75(3666)   (Show Source):
You can put this solution on YOUR website!
SEEING AS ONE SIDE IS ALREADY FENCED THE WE NEED 3 SIDES FOR 250 METERS.
X+2Y=250 OR X=250-2Y
XY=7500
(250-2Y)Y=7500
250Y-2Y^2=7500
2Y^2-250Y+7500=0
2(Y^2-125Y+3750)=0
2(Y-75)(Y-25)=0
Y-75=0
Y=75 METERS ANSWER FOR ONE SIDE
THUS THE OTHER SIDE IS:
250-2*75=250-150=100 METERS.
PROOF:
75*100=7,500

Answer by Fombitz(32388)   (Show Source):
You can put this solution on YOUR website!
OK, you have a rectangle of width, W, and length, L. You want a garden that has an area of 7500 square meters. The area of a rectangle is . Your first equation is :
1.
The total amount of fence you have to work with is 250 meters. But you also know that you don't need one side of the rectangle, one length or one width. Let's pick one length. So the perimeter of your rectangle is :
2.
The length is crossed out since there is already a fence present.Here are your two equations.
1.
2.
Use equation 1 to solve for L in terms of W and substitute into equation 2.
1.
Multiplicative inverse of W.

Now use the second equation and your result:
2.

Multiply both sides by W.

Additive inverse of 250W

Use the quadratic formula to solve for W.

where , , and .

and
if ,
2.

if
2.

Two solutions