a rectangualr field borders a river. A farmer has 750 ft of fencing to fence this field excluding the side the border the river. Express the area of the field he can fence in terms of one variable.
Lenght=Y (one side borders the river so no fencing is needed)
Width=X
2007-07-02 07:14:32 · 7 answers · asked by Girli3Dimpl3s 2 in Science & Mathematics ➔ Mathematics
A = LW
fencing = L + 2W
Substitute the second equation into the first to get the area as a function of W only:
fencing = L+2W
750 = L+2W
L = 750 - 2W
A = LW
A = (750-2W)W
A = 750W - 2W^2
Normally in this sort of problem, you go on to take the derivative and set it equal to zero to find the maximum area. But you only asked to place the equation for area in terms of one variable, and that's done above.
Note that the greatest area in this case is NOT a square (as another respondent suggested), as a result of the river boundary.
dA/dW = 0
d(750W - 2W^2)/dW = 0
750 - 4W = 0
W = 750/4 = 187.5
750 = L + 2W
750 = L + 2*187.5
750 - 375 = L
L = 375
The area of a perfect square (250 feet on each side including the river boundary) would be 250*250, which is 62,500 square feet.
However, the area for my answer (L=375, W=187.5) is 70,312.5 square feet, which is greater.
2007-07-02 07:17:39 · answer #1 · answered by McFate 7 · 0⤊ 0⤋
The greatest area is always a square, so normally you divide the available fence by 4 for equal sides. Since one side is the river, divide teh 750 by 3 to find the length of each side.
So each side is 750 / 3 = 250 feet.
Total area is 250 * 250 = 62,500 square feet
2007-07-02 14:18:48 · answer #2 · answered by therealchuckbales 5 · 0⤊ 0⤋
Y = side bordering river
X = other side
Fencing needed = 2X + Y = 750
2X = 750 - Y
Area = XY = 375Y - 0.5*Y^2
or 750X - 2X^2
If you wish to maximize this area
dA / dX = 750 - 4X
= 0 when X = 187.5
Y = 375
This max area = 70,312.5 sq ft
Note that it is NOT a square. It would have been a square had the fencing constraint covered the entire perimeter. Since it covers less than the entire perimeter, the problem changes a bit.
2007-07-02 14:19:11 · answer #3 · answered by Dr D 7 · 0⤊ 0⤋
Area = x.y
Perimeter = 2x + 2y
But one side (x) is on the river side and doesn't need fencing as per the given data. So, x + 2y = 750 ft.
x = 750 - 2y
Area = x.y = (750 - 2y). y = 750 y - 2y^2 sq.ft
2007-07-02 14:25:12 · answer #4 · answered by Swamy 7 · 0⤊ 0⤋
y=750-2x
therefore area = xy
=x(750-2x)
2007-07-02 14:25:38 · answer #5 · answered by ps 3 · 0⤊ 0⤋
if i am understanding the question correctly, A=(750-2l)(l)
l = Length
2007-07-02 14:22:04 · answer #6 · answered by Archmage 2 · 0⤊ 0⤋
1*748=748^2 ft....?
2007-07-02 14:30:38 · answer #7 · answered by Russh 2 · 0⤊ 0⤋