Respuesta :
This problem does not have solution.
When you do the algebra you find that the statementes lead to the equation of a hyperbola which does not have a minimum. And so there is not a minimum cost.
This is how you may get to that conclusion using math:
1) variables:
x: length of the side of the fence parallel to the road
y: length of side of the fence perpendicular to the road
2) area of the garden enclosed by the fence: xy
3) cost of the fence: multiply each length times its unit cost per foot
cost = 6x + 8x + 6y + 6y
cost = 14x + 12y
4) cost is also equal to $ dolars times the area = 2xy
So, 2xy = 14x + 12y
=> 2xy = 14x + 12y
Also, do not forget that x and y has to satisfy x>0 and y>0
You can solve for y (or x if you prefer)
2xy-12y=14x
xy - 6y = 7x
y(x-6)=7x
y = 7x / (x -6)
You can verify that as you increase x (starting any x > 6 to make y positive) to make y minimal, y will decrease but the produc xy will increase.
for example do x = 100, you get xy ≈ 744, x = 1000 xy≈7042, and that trend never ends.
If you know about limits you can show that.
At the end, there is not a minimum cost
When you do the algebra you find that the statementes lead to the equation of a hyperbola which does not have a minimum. And so there is not a minimum cost.
This is how you may get to that conclusion using math:
1) variables:
x: length of the side of the fence parallel to the road
y: length of side of the fence perpendicular to the road
2) area of the garden enclosed by the fence: xy
3) cost of the fence: multiply each length times its unit cost per foot
cost = 6x + 8x + 6y + 6y
cost = 14x + 12y
4) cost is also equal to $ dolars times the area = 2xy
So, 2xy = 14x + 12y
=> 2xy = 14x + 12y
Also, do not forget that x and y has to satisfy x>0 and y>0
You can solve for y (or x if you prefer)
2xy-12y=14x
xy - 6y = 7x
y(x-6)=7x
y = 7x / (x -6)
You can verify that as you increase x (starting any x > 6 to make y positive) to make y minimal, y will decrease but the produc xy will increase.
for example do x = 100, you get xy ≈ 744, x = 1000 xy≈7042, and that trend never ends.
If you know about limits you can show that.
At the end, there is not a minimum cost
Answer:
There is no minimum cost for the fencing that can satisfy these conditions.
as the equation obtained for this problem is y = [tex]\frac{7x}{x-6}[/tex].
Step-by-step explanation:
let length of the rectangular rose garden be x
and breath of the rectangular rose garden be y
then the area of the garden = xy
cost of fencing a foot for the 3 non road sides = $6
and cost of fencing a foot for a road side = $8
total cost of fencing = 6x + 8x + 6y + 6y
so, total cost = 14x + 12y
cost of fencing for every square foot of the area =$2
So, 2xy = 14x + 12y
=> 2xy = 14x + 12y
Also, x and y has to satisfy x>0 and y>0
You can solve for y (or x if you prefer)
2xy-12y=14x
xy - 6y = 7x
y(x-6)=7x
y = [tex]\frac{7x}{x-6}[/tex]
we can check that this function has no minimum value.
as we increase x (starting any x > 6 to make y positive) to make y minimal, y will decrease but the product xy will increase.
So, there is not a minimum cost for fencing that can satisfy these conditions.
